This is a complicated question to ask unless related to a specific part, but it asked very often.  Some parts you should probably never buy used.  Some used parts can be better than rebuilt parts.  Some new parts are not as good as used parts... Don't worry, I'll explain!

The first rule is the easiest... never buy used rubber consumable items.  Even if the car had low miles, rubber tends to "dry rot" over time, and even a new looking part may be years old and prone to fail.  Rubber items are some of the most important items on your vehicle, and you do not want them to fail.  This suggestion covers radiator hoses, heater hoses, belts, vacuum hoses, wiper blades, and especially brake lines. Of course, tires are rubber, but the same can apply.  If you notice, most places that sell tires will not offer a warranty, and its for a reason - the liability is too great!  If you do decide to purchase used tires, check the inside for foreign objects, patches, and plugs.  Look at the "bead" area (where the rubber meets the wheel) for rips and/or tears.  Squish the tire down and look for fine cracking which indicates dry rot. Used off-road tires are less likely to have dangerous results from a blow-out, but still beware.

Rule two: Avoid remanufactured electrical parts from unknown sources and/or budget auto parts chains.  This is especially true of starters, alternators, etc.  Not many people realize that most of the places only replace the one item that failed in the assembly.  So, if you get a rebuilt alternator that had 200,000 miles on it, they may have only replaced one bearing (or even just cleaned and painted it), and the rest of the part is worn out and ready to fail.  On the other hand, a good used part will have low miles and be much closer to new condition and a fraction of the price of new.  Expect to pay from 10% to 50% of new price for a good used alternative.  If you have the money to spare, buy new, but good used items are a great alternative.

Rule three:  Body parts or sheet metal can be tricky.  Of all the parts on your car besides wheels, the sheet metal makes or breaks the looks of your ride.  Ill fitting body parts can make even a $100,000 car look bad.  The main thing to consider in body parts is OEM (original equipment manufacturer) fitment.  New OEM panels fill like they were made from the car.  They can be expensive.  Used panels often time are OEM, but make sure before purchasing.  Used parts can be great because many times you can actually find them painted the same color as your vehicle.  Make sure to check for small dings and previous repairs though.  Plus, sometimes you can get cheaper prices for buying more parts like a "front clip" for example.  Aftermarket parts is where it gets tricky.  Sometimes they are really cheap, which sounds great, but they may not fit, mounting holes may have been omitted, or they may be really thin and easy to dent.  When buying aftermarket sheet metal, look for the CAPA certification sticker (don't just take the sellers word for it), or if it is from Keystone (one of the largest distributors of aftermarket sheet metal), look for the Platinum Plus sticker - it is guaranteed to be as good as original.

Rule four:  Low mile engines aren't always best.  This sounds totally wrong, but it is true.  Engines history is just as important.  Of course a new engine from the manufacturer is going to be just like new off the lot, but the price can be very high!  Used engines can often be much cheaper than even a rebuilt engine, but it can be a toss up as to which is best.  Low mile engines are generally the best way to go, but ask about warranty and where they came from.  I state this for one reason - Japanese engine swaps!  Many places advertise ultra-low mile engines from Japan.  This sounds great except most people do not understand Japanese emission laws.  New JDM (Japanese Domestic Market - we are USDM) cars have to pass a very strict emissions test somewhere around 30,000 miles.  Many well maintained cars can't even pass it.  SO, some people do not even change there oil in the first 30,000 miles, and just get rid of the car come inspection time.  Doesn't seem like as good to buy a low mile engine from Japan now.  Don't get me wrong, you can get good maintained engines, but beware.  Let us not forget about rebuilt engines.  If you know the rebuilders, or if it is a reputable rebuilder company like Jasper, you will get a good engine.  But beware of some auto part chain's rebuilt engines.  I have heard of many rebuilt engines from one particular parts house with parts from A to Z that have blown or started smoking within a couple thousand miles.  The same goes true with used engines; the reputation of the dismantler/salvage yard means a lot (kinda like Ebay!).

Rule five:  Shocks and struts can't be trusted after about 50,000 miles.  They go bad.  If you car has 60,000 miles on it, and a strut goes bad, why would you replace it with one with 40 or 50,000 miles.  It just doesn't make sense.  Unless its low mile, or you can't get it through an auto parts store - don't buy used.

Rule six: Used suspension parts are your friend.  Most rigid suspension parts rarely fail unless damaged in an accident, therefore its almost always cheaper to purchased used parts.  Just beware of worn bushings.

Rule seven:  BUY NEW MOTOR MOUNTS!  Don't waste your time or the salvage yards time... they are cheap at parts stores.  The only exception is discontinued mounts that will have to be sourced used.

Rule eight:  Install seals when installing engines and/or transmissions.  This is $10 well spent, plus most salvage yards will not warranty seals or the damage they cause when they fail.  I can not say this enough, install rear main seals and front pump seals!

Rule nine:  Explore all wheel options.  Don't buy new replacement wheels from the dealer.  You can get quality used wheels at salvage yards, you can get yours fixed for about $100-150 each, or you can buy reconditioned wheels through many salvage yards that are as good as new for less than half-price.  Also, don't forget, sometimes is just as cheap to buy a set of new aftermarket wheels, or a different set of used wheels!

Rule ten:  Use your head and the internet!  Don't just do or buy the first thing you see - its that simple.  Think about your options, used the net and/or Ebay to see what options you have.  Call your local parts house and or salvage yard.  You can save a lot of money just by wasting a little time doing research!

I hope this helped.  I own HWY 79 Auto Sales & Salvage in Caneyville, Kentucky.  My family has been in the automotive industry for over 35 years (before I was even born!).  We have owned three parts houses and this salvage yard.  I have a mechanical engineering degree form GMI/Kettering with a specialty in machine design, so I think I have a good idea of what I'm saying!  If this was helpful, let me know and I'll try to post up more guides.  Thanks!