You are looking for a simple digital camera.
Easier said  than done, right?
That's why I'm here to help you.

I suspect you are just looking to upgrade yourself into the digital age and are saying "I need a digital camera!" You're not a portrait photographer, you aren't a photo journalist, you are just Regular Guy/Gal looking to take pictures at family gatherings, of your cute puppy, your kids, etc...

I've been in the photography biz for many years, photographer and commercial labs.  When it came to researching a simple point and shoot digital camera I had to wade through tons of types, models, reviews and information.

Let's begin: (don't worry, you can skip the confusing high-tech words and go straight down to the camera's I've got listed for you)

"What is a megapixel" - pixels are the tiny dots that comprise an image, if you aren't a pro you don't need to know more than that! Did you need to know how much silver was in the film you used to buy? No way, so don't worry about it - a 4 or 5 megapixel (MP) camera is going to be fine for pictures enlarged up to 8x10. 
You can always look online for a more in depth explanation. :)

Optical zoom - know what that is!? A zoom lens, just like on a 35mm point and shoot camera!  Optical or digital zoom, don't fret - it's just a fancy term for zoom.  Unless you're taking super close ups (or macros) for scientific photography you don't need to worry about what magnification you are getting.

LCD - liquid crystal display, also known as Color Display - that's the square thing on the back of the camera that allows you to see what you are taking a picture of  - or play back images.  It can be large or small. It replaces a Viewfinder, the small thing you used to stick your eye up to in order to see what you were taking a picture of, and if you are like me - you will stick your eye into the LCD and get it all smudgy till you remember you don't need to do that.

Flash - yes, you need one and all point and shoots come with it.  More than not you will have options for red eye reduction as well, but that can be annoying and frankly I advise turning it off.

ISO - stands for International Standard Organization, it used to rate the speed of film.  In digital cam's it's fake, in a way, but an easy way to know what is what: 100 - 200 good for daylight, 200 - 400 good for indoor, 800 - 1600 night shots.

Shutter speed - this tells the lens how long to stay open, keeping it open longer will let in more light, but the numbers are confusing.  1/1000 is for super sunny days, 1/60 is for flash, 1/30 long exposures.  If you are into simple - you don't even have to worry about it.
Many simpler camera's come with little pictures of a sun or moon, etc., making it nice and easy to pick the setting, called Scene Modes:

Scene mode: little pictures built into your camera that show through the LCD that help you choose what type of exposure you ought to use. Auto, portrait, landscape, close up, sports, night portrait, snow, beach, self portrait, party, fireworks, backlighting, kids and pets, etc... It does the work for you, just choose what you are intending to shoot.

Image - new term for picture, or shot.

Memory or Storage - like film you have to store your images somewhere, now with a plastic little 'card' that stores all your images. You'll need to know type: SD, xD, Compact Flash, to name a few. 
And Size: You will want one that holds a lot so 256MB or bigger. You may get a 16MB card with your camera  but it's only good for about 10 shots.
TIP: Be sure you ask whoever it is you are buying from what type of card you need - because not all will fit all camera's. Also, if a USB comes with the camera or if you will need a card reader.

Docking Station - a small printer to put your camera on to in order to make prints.

USB - a cable that runs from your computer to your camera to let you transfer images from cam to computer.

Card Reader - a device that you put your memory card in to and then that plugs in to your computer for downloading.

Upload (or download) - this is to transfer the images that are on your card to a different place, either your computer, a docking station, etc...

Batteries - also called Power Supply, I say stay with 2 AA types, as it's easier and less expensive.

You've got a few details above, you are also going to run into lots of options that you aren't going to give a hoot about.   Movies for one - and since I'm trying to keep it simple I'm not even going to go into the movie taking!

Tip:  write down a few of the options you are going to use - like outdoor setting or indoor with flash, and keep it with you - that way you can make the adjustment on the go and not worry about goofing up the camera.

I'm just listing camera's that are easy and are under $200 for you, to give you some ideas, not all of their options are listed here, just some highlights. They all come with flash, zoom, wide, viewfinders and LCD's. They all allow you to play back your pictures and delete if you wish. Each comes with a mount on the bottom for a tripod, always useful and I'm sure you can find a great deal on Ebay for a small tripod!

List price is from the manufacturer, where I could find it, low price is what I found online.

Tip: I recommend a viewfinder as well, not all camera's come with them, but on a super sunny day you are not going to be able to see the image on the LCD panel, no matter what the manufacturer tells you!

Camera pictured under description:

Kodak PowerShot C533 list price $149, online low price $115 :  5 MP, 5X digital zoom, 13 scene modes, easy video, undo image option, memory plus with internal memory, also works with Easyshare docking station.

Kodak Easy Share series: These come with a red button that tags (chooses) a shot that you want to print or email later.  Also has the feature which allows you to place the camera onto the printing docking station, one button and your 4x6 print pops out.

C310 List $109, low $81: 4MP, 5x zoom, fast image capture, quicktime movie


C330 List $149, low $97: 4MP, 5x zoom, 13 easy access scene modes


C530 List $119, low $92: 5MP, 5x zoom, no delay image capture

Canon A430 List $149, low $113: 4MP, 4x zoom, macro lens (that means super close up), low power usage

Fuji FinePix A400, low $104: 4.1MP, 3x zoom, large LCD


Fuji FinePix A500, low $117: 5.1MP, 5.2x zoom, large LCD

Next - take this list to a dealer, either a camera dealer or superstore.  You want to get the feel of the camera, not just read about it.  Some seem bulky, to me the camera's with a viewfinder are a little bulkier but if that means I can see what I'm shooting no matter what - it's worth it!
Do the controls and features seem simple enough?  Are there too many features?  Write down your notes, you'll need them when comparing.

If shopping online, still go to the store and get an idea, also take a look at the pictures OF the camera's online, you may be able to weed some out just by the easy look of the controls on the back of the camera.

You do all this and still, you think "smaller is better" - I'll tell you, I was going to buy the Nikon Coolpix L3.
It is sleek, easy to use and tiny, but I didn't like the LCD (used my friends cam in Florida, a good test of sunny conditions), it's a little pricey and the website very annoying.

My second non-viewfinder cam was the Olympus FE 110
, good price and options but difficult to find in the stores to look at, but great website.

Other: some camera's come bundled with a strap, batteries, a low MB memory card, instructions and CD.  I would 'guess' that all the camera's come with them, but be sure to ask or make sure what it comes with.  Any reputable Ebay dealer or store will be happy to answer questions before you purchase or bid on an item.

Warranty's and extended warranties: that's your decision.  I bought my cam online and if there is a problem I will first go to whom I bought it from, and then if that fails I will go to the manufacturer.

Refurbished: this means used, don't do it.

I hope this has helped you a little and saved you from the frustration of beginning your search all by yourself. 
Remember, don't get anything more than 5 MP, zoom is just zoom, a viewfinder can really come in handy, and AA batteries are easier to go with.