I wouldn't want to recommend a digital camera for anyone without knowing a lot more than I do but I have some experience with several. First of all there's no doubt that they win hands down over film for usability, features, cost, flexibility, you name it. But there are things you don't necessarily think about until you've cursed at the thing a few times.
Viewfinder!!!! I simply can't use a camera if it hasn't got one. Many haven't.
How long does it take from power on to taking the first picture? I've missed a lot of shots because my (seven year old) Fuji takes forever to go through its power up routine. It's not so much point and shoot as press ON button, point, wait, curse, and shoot if it's still there.
Optical zoom? The reason I bought the Fuji. It's excellent but there are better now.
Image stabilization? Well worth the price IMO.
Primary cells or rechargeable? My main gripe about the Fuji is poor battery life, and it can't use rechargeable types.
Battery availability in remote areas? AA cells are everywhere, others may not be.
Type of memory card, size, cost, availability? The SM cards in my Fuji are almost obsolete and the biggest you can get is 128Mbyte. My wife's two year old SLR takes up to 16G and they're cheaper!
Related to memory is the ability to take images at reduced resolution so you can get a lot more shots on a card if you want to. I can get about 100 low resolution shots on the Fuji's supplied card, my wife can get several thousand on hers. She took nearly five hundred one afternoon at an air show and some of the results were spectacular.
If you're snap-happy and take thousands of shots then discard the rubbish you'll get some great results BUT it will take most of the rest of your life if you aren't careful!
Movies? Some can't do it, probably not many now.
Remote control/remote flash? Getting into SLR territory perhaps.
I'll leave out obvious things like size and weight, you'll know it when you see it.
Definitely try out a few before getting one or you might regret it. The first one I bought for my wife's birthday went back to the shop within a week. She's still got the second one but it doesn't see much action now she has an SLR.
If you want stunning pictures I'd say go for an SLR if you can stand the price and the weight and bulk (don't forget the extra lenses), but if you just want snaps then most point & shoot things available now will be well up to the job.
And get a bigger disc drive for your computer.

She now has over 20,000 shots. So she wrote a little 'picture of the day' program so that she sees some of them, otherwise they'd just gather metaphorical dust on the disc. It's great to get a nice surprise when you fire up the browser.