I have a Nikon D50 (precedes the smaller D40) and it takes awesome photos. Downside is lugging a bigger camera with you. This one won't go in a shirt pocket, but you can't take better photos with a Point-and-shoot, period. Coupled with the 55-135 zoom lens with image stabilization, it will turn any hack into a good photographer. With the IS turned on, I can shoot at max zoom without a tripod and don't get any blurriness. I bought it mostly for taking action shots of the little ones tearing around as it can fire 5 frames per second (non-flash). It wasn't cheap, but I've seen some good deals on the D40 lately so it may be worth a look. You won't be disappointed in the image quality.
I also have a Panasonic Lumix L-75 (I think that's the model), which is point and shoot, runs on AA batteries, and cost me $99 a year and a half ago. The downside is it gets grainy and blurry in low-light situations, and the flash tends to wash things out. Good little budget camera, but it won't win me any photo contests.
My wife just bought a Kodak Easy share 913 (I think, I stink at remembering multiple model numbers), and she says it works great. Proprietary battery, which all the small ones have to save space and weight, lots of mega pixels, and small. Haven't taken enough photos to render judgment on low light or action shots. For $123 at Wal-Mart (with some extra goodies like a case and lens cloth) the price is hard to beat.
Megapixels have reached their useful capacity as a quality indicator long ago. Anything over 6mp will be enough for anyone as long as you print to 8x10 or smaller. Bigger photos may be an issue at 6mp and less, but I gather most here aren't concerned with large prints.
Good luck with whatever you choose. Just keep in mind the useful life of these things gets shorter every year. If you get more than 3 years out of one, consider yourself lucky. As Jay suggested, go for good quality lens with OPTICAL zoom (not digital zoom), and look into image stabilization. It will eliminate camera shake for those zoom shots.
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