It's going to be different requirements for different people, depending on how the camera is to be used...so to some degree you need to keep your own list of what you need and how you intend to use it. However, seeing what other people used to base their purchase can help get ideas. So in that vein...
What I was looking for in a P&S camera was a 2nd camera to go with my DSLR - so mine wasn't to be a primary camera. My most important features/abilities were:
1. Super-duper compactness - smallest I could get. I wanted to lose it in a pocket and not even know it was there.
2. Nice screen and decent battery life. It didn't have to take 3,000 shots on a charge, but last all day for casual snaps.
3. Maximum flexibility in various lighting situations - a very hard requirement for P&S, but something I wanted to at least look for.
4. Stylish and well built. Sure, noone else likes to mention that looks have something to do with it - but I can be straightforward...I like brushed metal, high style, cool colors. It's a second camera, so I want fun and eyecatching.
5. A few key controls I cannot live without in a P&S - spot meter, spot focus, EV control, optical stabilization. I don't expect manual control in a compact like I want...so I reduce my expectations and make sure I at least have those things.
I ended up with a Sony TX1 - it fit the most requirements. It's ultra-super-duper small. It's all-metal, internal zoom, sliding lens cover, blue-bodied, with a huge touch screen LCD. It has spot focus, spot meter, EV, and optical stabilization. With it, I can shoot in anything from good light to nighttime handheld - extremely rare for a compact. Battery is good enough to get me through a day, and was upgradable to a nicer Infolithium model, which gives me battery life remaining in percentage right on the screen - much better than the silly 4-bar indicators.
Had I been looking for a primary camera, and compactness was not quite the same priority, I would have gone a totally different direction. But my DSLR does primary duty and is uncompact, so I needed an ultracompact!