In 1984 or so, it was thought that the analog synthesizer would be a thing of the past.  However, this great device has had a building resurgence since the mid 90s that has risen to the level of mania!
    eBay has had a big influence on this craze, being the best source of analog devices.
    eBay may be the best thing to happen to the analog synthesizer enthusiast, but it is not without pitfall.  There are issues of misleading auction pages, synthesizers with unstated problems, sellers who are not aware of the operation or disfunction of their devices, etc.
    There are a lot of people who are just entering into the field of analog synthesizer usage (and especially analog synthesizer purchase on eBay) who may not be aware of some of the facts about the synthesizers themselves, how they work, what to look for, and what to avoid.  These people may be looking for advice on what to look for in an auction to at least increase their chances of getting a decent, functional synth at a reasonable price.
    That's what this guide is all about!
    What is analog?  Basically, analog sound is sound where the frequency of electrical impulses that exist inside the wire are equal to the frequencies of soundwaves that are produced when those impulses are translated to soundwaves by a speaker.  The sound inside the device is equal to the sound outside the device.  As a result, the sounds generated by devices that are analog are more pleasing to the human ear, being essentially equal to acoustic sound.
    Digital devices create sound using mathematical models that are then translated into analog sound.  The resultant sounds are very clean, clear, and noise free... but have a sterile, unnatural quality that is not as immediately pleasing to the ear.  Digital soundwaves, before translation, are not equal to acoustic soundwaves, but rather soundwaves that are made up of individual points that are connected together by your ear... like a soundwave dot-to-dot representation.  The higher sampling rate (those little dots are called samples), the more the wave sounds acoustic... but it's only a fabrication that can kind of fool your ear.  The differences are minor, but detectable... or people wouldn't favor analog!
    Want to buy an analog synth?  Let's get started:

1.  Study the Models
    Just starting out in the world of analog synthesizers?  The best way for you to achieve success in your quest is to know the various makes and models of analog synthesizer.
First, find and read the book Vintage Synthesizers by Mark Vail (there is likely a copy of it available right now on eBay!).  It is packed with information, history, and pictures of various analog synthesizers and their makers.  It is a great place to start, and a resource that will allow you to ascertain whether the websites you visit are factual or hype-based.
    Then, visit websites like www.vintagesynths.com, synthmuseum.com, and sonicstate.com.  These websites contain a lot of information about the various models of analog synthesizer.  Sadly, all of them are lacking in some way or another, and all of them (especially vintagesynths.com) contain falsehoods and errors.  So, keep an open mind and compare the information you read against other information you read.
    A truly great resource is the archives at Analogue Heaven.  search.retrosynth.com/ah/ is the place to go, enter in the name of the synth you are interested in, and you will have access to information about that synth discussed by professionals and amateurs alike.
    On eBay, enter keywords like "analog synthesizer," "analog synth," "vintage synthesizer," "vintage synth," "analog keyboard," etc.  One of the best ways to begin to familiarize yourself with models of synthesizer you might be interested in is to study the market, what is available, and the pictures the seller posts.
    The more you know about the individual models and their functionality, the more you will be able to decide what you want, what synthesizers are truly valuable, and what system will work best with your intent.


2. Study the Lingo

VCO:  Voltage controlled oscillator.  The component that generates tones.  
Square wave:  waveform that has a hollow tone, modulation that jumps between two levels
Saw wave: waveform that is very bright and brassy, modulation that ramps up and drops
Triangle wave: waveform that is dark and pure, modulation that sounds like a sine wave
Sine wave: even warm pure tone, modulation that basically goes up and down
Pulse wave: a rectangular-looking square wave with a nasal quality
Pulse width modulation: a waveform that varies between a square and a pulse wave.  Creates a natural chorus effect.
VCF:  Voltage controlled filter.  Usually a low pass filter that increasingly removes high frequencies.  Sometimes also a     High Pass.
Filter Cutoff point:  The point where frequencies begin to be subtracted by the VCF.
Resonance/Emphasis: A filter feature that amplifies the frequencies that exist just before the filter cutoff point.  A mainstay of analog sound.  Responsible for the "wah" sound.
Self-oscillation:  Usually a feature of 24 dB per octave filters, this is when the resonance is so high that the filter starts to    generate a sine wave whose frequency is defined by the filter cutoff point.
ENV: An envelope is a list of voltages that usually define the volume of the sound output by the VCA (does the sound have a slow or fast attack?  A slow or fast release, etc), or the filter cutoff point in the VCF.
VCA:  Voltage controlled amplitude or amplifier.  This is what is in charge of the volume of the sound, and what shape that sound will have.
LFO:  Low Frequency Oscillator.  This oscillator's function is usually to modulate the VCO, VCF, or VCA, creating vibrato, growl, etc.
Sample and Hold: a modulation that varies between random voltages, creating random pitches or cutoff points.
CV: Control voltage... voltage used to control the various components of the synthesizer.
Portamento: Control voltage effect that results in a "slide" between pitches.  Essentially glissando.

3. Study the Features
    Before you buy an analog synth, you must know what features are available on analog synths, and which of those features the model you desire possesses!  It's important because auction price, Buy-It-Now, and ending prices may not reflect functionality or features at all.  Often, synthesizers go for the price they do because there is a hype associated with its name, and this hype may have nothing at all to do with what the synthesizer can do.
    Examples of this include synthesizers like the Roland SH-101, or the TB-303.  These synthesizers are not particularly feature-filled, but have a cult status generated by genres that depend heavily on those models.  They have a characteristic sound that is desireable by many, but are comparatively limited in their features.
Features to look for:
How many oscillators does the synth have?  The more oscillators, the better.
How many waveforms can the oscillators make?  More waveforms means more diversity in sound.
Is the filter 24 db/oct?  Filters usually come in 24 dB or 12 dB.  Both are great, but 24 dB will likely have a more desireable sound and the ability to generate filter oscillation.
How many envelope generators does the synth have?  Many have one that will control both the VCA and the VCF... more desireable is when there are separate ENVs to control the VCA and VCF.
How many stages does the ENV have?  Is it an Attack/Release envelope, an Attack, Sustain, Release envelope, or an Attack, Decay, Sustain, Release envelope?  ADSR is best!
How many waveforms does the LFO have?  More is better!
How many control voltage and output ports does the synthesizer have?  

4. Study the Market
    Enter the keywords listed above, and any others you can think of that relate, as often as possible.  Study the information portrayed in the auctions.  Note what the sellers are asking and eventually what they get.
    Many eBay analog synth sellers don't know what they have and do not price their item accordingly.  I have seen perfectly worthless synthesizers with laughable prices, and extremely valuable synthesizers with extremely low prices.  This occurs when a seller doesn't do research on the model of synthesizer they are selling before they post it.
    When you study the market, you will start to see pricing trends of the various models, which will help you decide what you want and what you can afford.
"Rare," "vintage," and "analog" are all terms that are abused.  If you want to have fun with listings, look at what sellers are selling with these terms.  "Rare" is often applied to synthesizers of which there are several others selling at the same time.  "Vintage" is often applied to anything previous to 2000.  "Analog" is applied to devices which may have analog components somewhere in it, but are not considered analog at all.  For example, the Ensoniq ESQ-1 is a great older synthesizer with an analog filter, but it is SO DIGITAL in so many respects... it uses digital and sampled waveforms!  When a person is looking for "analog" they are usually not looking for digital synthesizers with analog aspects... and if they are, they might just as well look for digital synths or analog emulative synths.


5. Study the Dangers
a.  "I couldn't test this."  "This is untested."  Are you a gambler at heart?  Well, this is a bad bet.  Sure, it's possible that they haven't tested it and/or cannot test it, but this is often code for "I know this is broken, but hope I can sell it anyway."
b. "I played with the knobs and everything seemed to change the sound in some way."  I used to get really excited about this one... it seems like proof that the thing is working, right?  Well, I've been stung by this one several times.  There are PLENTY of times when a very broken synthesizer's knobs or sliders will change the sound.  The question is, do they change the sound the way they are SUPPOSED to?
c.  "A friend who plays keyboard checked it out..."  "A friend who is a musician checked it out..." "A friend who is a guitarist checked it out..."  "A friend who knew a guy who knew what Moog meant checked it out..." "A friend who has Moog brand auto parts in his truck checked it out..."  This one hasn't worked out well for me, either, unfortunately.  If the seller can't tell you if it works or if it works well, you are taking a risk.
d.  "The pots are a little scratchy."  This usually isn't a very big deal... but sometimes it is.  Scratchy pots are really common.  Every pot gets scratchy eventually!  Sometimes working the knob fixes it.  Using a conductive lubricant can also fix it, but they're often advised against by experienced techs.  I wish I knew how they actually take apart the pots and clean them, but I don't!  Anyway... when the scratchy pot is the  on/off volume knob, or sometimes just the volume knob, it means that sometimes that dirty pot can cause the synth to cut out!
e.  "I'm sure it works, it worked the last time I used it."  This usually translates to "This doesn't work, but I want to sell it anyway."  Or, "I think there may be something wrong with it."  Or, "I don't want to turn this on and find out that the 5 years it's been in the attic has had a deleterious effect upon it."  Always encourage sellers to TEST THE DANGED THING.
f.  Research the model of synth you're looking at to find out if it has specific problems you can ask the seller about... like the black goo in Moog Opus 3s and Realistic/Moog MG-1s, the battery leak problem in Korg Polysixes, the connector problem in Moog Sonic Sixes, the connector problems in Korg PS-3100s, the millions of problems with Polymoogs, etc.

6.  Study the Seller
A good analog synth seller should list a lot of information about the functionality of the synth.  They should feature a fair amount of clear pictures of the item being sold.  They should be forthright and relatively quick to answer questions.  Their auction should list a lot of their personal experience with the device.  Their auction should be very specific about the condition of the device.
A questionable seller is one who basically cuts and pastes the boring and inaccurate posts from synth websites and uses this information as their own.  A questionable seller won't answer your questions.  A questionable seller has dark, blurry pictures.
It's always sad when the picture in the auction isn't a picture of the actual device.  I've seen this happen a lot... but it does not definitively portray the seller as good or questionable.  When the seller is using a picture that is not of the device, I would be wary if they don't make up for this with lots of friendly question-answering e-mails.
ASK QUESTIONS.  Ask each seller every question you can think of.  They want to sell the device, they should be happy to answer even the most boring or dumb question.
Check their feedback.  Feedback is a funny thing on eBay... I've heard a thousand times "don't deal with anyone under 98%!"  Which is silly.  I have been involved in eBay auctions that could have gone very badly.  Sometimes bad people leave bad feedback... and there isn't much you can do about it.  Look to see if the bad feedback the seller has forms a TREND... if you find a couple of negative feedback posts that are in the same vein, like "seller took forever to ship," or "didn't answer e-mail," etc.  Those are the things to watch out for.  
A BIG thing to watch for is complaints about packing.  Packing is the single most important aspect of online synthesizer buying.  Before you bid on a synth, you should have a very candid discussion with the seller about how they plan to pack the synth.  It is not worth bidding on the BEST SYNTH DEAL EVER if the person says "well, I'm going to put it in a box and send it USPS."  Your mint-condition Musonics Minimoog is not a great deal at $1500 if your seller plans to put it in a too-large thin-sided box with other pieces of cardboard as packing material.
Ideally, an analog synthesizer should be packed like this:
The synthesizer should be SEALED in a plastic bag, or well-taped bubblewrap.  You do not want to expose it to any moisture it may meet on its journey.  More important than that, you do NOT want bits of packing peanuts/popcorn making their way into your synthesizer.  Ideally, your plastic/bubblewrap wrapped synth could then be encased in a layer of cardboard (this step isn't necessary, but is helpful).  Then, the synthesizer should have at least 4 inches of packing peanuts or popcorn, and then a sturdy well-sealed box.  The heavier the synth is, the more popcorn it will need.
I have received synthesizers that were drowning in a sea of popcorn, and synthesizers that seriously only had one layer of cardboard between its switches and UPS Ground.  I have been relatively lucky.  A synthesizer CAN survive bad packing, but you don't want to take the risk.

7.  Study the Hype
Some analog synthesizers fetch high prices because they have followings!  Whether the reason is a specific sound, a specific function, or a genre/band which uses that synthesizer, it can be very frustrating when a synthesizer you like or want for personal or practical reasons has a following that will pay through the nose for it.  
Just because a synthesizer is a Moog, or it used in your favorite genre, doesn't mean it's going to be a synthesizer that suits your needs.  I can think of a few Moogs that are not great at all, and some of the synthesizers that are known for their use in certain genres are complete crap.  Don't buy a synthesizer simply because it is expensive, or because it is famous!  Instead, try to find a synthesizer that suits your personality and needs.
The flipside of this is that there are synthesizers out there that are ignored or reviled for absolutely no reason.  Several synthesizers I have purchased were advised against by really lame websites I won't mention... which I discovered were actually FANTASTIC analog synthesizers with great sound and features!
It's best to avoid the hype and look at the objective truth about features.   If you're going to use "sound" as your deciding factor, look to the professionals!  A synthesizer that sounds good because it was  used in a song that everyone likes is not the same as a synthesizer that sounds good and is used by many professionals across a wealth of genres.

I hope this guide provides you with some things to think about!  If I can, I plan to edit it... adding things I think of later.  There is a lot to think about and know in regard to the purchase of an analog synthesizer on eBay!