I'm partial to Nikon myself, my SLR, DSLR and my point-n-shoot are all Nikon. I personally find that Nikon has better in-camera stability, which is great so you don't have to have a tripod. I like their point-and-shoots because they are SUPER durable--I've dropped mine a ton and still takes great pictures, no need for any tuneups, no problems with lenses or anything. Also I think they are easy to learn how to use, but they also supply a lot of advanced features for if you are more into photography. I love that I can choose my aperture and things like that on my point-and-shoot...so I don't always have to take my d70 with me.
That being said, my sister has a Pasonic Lumix and she really likes it. I think its a bit bulky, less features, but pretty good picture quality. I really dislike my grandpa's Fuji---I just don't think it takes as good of pictures for the money as some others, and I find it annoying to use with a very touchy autofocus. But maybe its just that model.
What you need to look for is optical zoom (not digital zoom---that is essentially cropping the photo in camera), your file capabilities (size, megapixels, etc) and the compatibility for memory cards and battery. Rechargable Li-Poly batteries are the most powerful and last longest than anything that takes double As or even the fat c2 (i think) kind.