I love my Canon PowerShot S70! http://consumer.usa.canon.com/ir/controller?act=ModelDetailAct&fcategoryid=144&modelid=10460. I love the quality of the pics it takes and the software it comes with is really easy to use. All the features make it look a little intimidating, but after a quick browsing of the manual it's really user friendly. Downside of the camera is that it is a bit larger than some of the really cute tiny ones out there, but I chose power over appearance.
Things I would look for in a cam: Resolution. The higher, the better. Though, the higher, the more expensive. I would say get at least 4 or 5 megapixels for a good quality picture. Go higher if your price range allows though.
Auto focus: Make sure it's good. Some cameras have poor autofocus, which can get annoying. It's hard to tell on the LCD screen whether the image is focused well.
And my personal pet peeve: Battery life. A lot of cams come with proprietary batteries that are expensive to replace and hard to find.
I have a Canon Powershot A95 and love it! It takes great pictures especially outdoors. I've never had a problem with it. The only complaint I have is that it takes 4 double A batteries and they don't last long so I have gotten 3 sets of rechargeable batteries and always have one set on charge, one set in the camera, and one set in my purse as a spare.
My brother has a Sony that is a few years old and he's had lots of problems with it. It is a cybershot. I hear other people don't have problems with them but from my brother's experiences I can't recommend them.
My rule of thumb for cameras is to buy from a company known for making cameras. Canon, Nikon, Olympus.
High megapixels make your pictures come out sharper and clearer which is what you want. Also pay attention to the Zoom. You want higher optical zoom and don't pay attention to digital zoom because your pics can come out really badly when you use digital zoom.
CC has a Canon Elph and hers takes really good pics too. Not sure the model number but you maybe able to search back posts to find out.