The most common question I get when I meet new photographers is, "which [compact] camera should I buy?" My answer these days is the Fujifilm FinePix F30 or F31d. It's the only compact camera that shoots decent images at high ISOs, which means that you can capture an image primarily lit by ambient light even in low(er) light conditions. The F30 and F31d produce great images at ISO 400 and usable images (debatable) at ISO 800 and 1600.
In comparison, nearly every other compact camera out there produces usable images only at ISO 100. At ISO 400 on those cameras, the images are nearly unusable. It's too bad, because I like the form factor, interface, and (non-image-quality) performance of some other compacts better.
Here's a situation that also comes up frequently:
Eric pulls out Fuji F30.
Random dude looks at camera and says, "Oh! You have an old camera." (as we know, thicker = older)
I don't reply.
Random dude pulls out a Casio EXILIM/other thin camera and says, "Here's mine. Look how small it is."
I feel like smacking him.
DPReview has posted an excellent article called Compact Camera High ISO Modes. It's worth reading if you're like the dude above, or if you're someone who thinks a 10 megapixel compact camera is better than a 4 megapixel digital SLR. Also, check out the DPReview F30 review.