I say this as a bit of a Nikon SLR fan-boy...

I think the best P&S cameras are the Canon PowerShot S series. They have about the same sensor and lens as the big and bulky G series but in a nice compact package. To keep budget, I would pick a generation old PowerShot S over most current P&S cameras. I don't know of other brands that make comparable cameras, so feel free to search and compare...

Looking at the latest PowerShot S110, here are what I consider most important specs:

• Up to F/2 lens
• 1/1.7-inch CMOS sensor
• Good zoom range that gets wide: 24-120mm (35mm film equivalent)
• 3 inch screen
• 1-1/2000 sec. shutter

I again say this as someone who owns a big SLR - size is important. To quote Chase Jarvis: "The best camera is the one that's with you." So pick out something you don't mind schlepping around.

There are 3 different variables the camera can manipulate to get a good exposure: shutter speed, aperture, and ISO. Different digital cameras have varying abilities with regards to aperture and ISO.

The lens (aperture): F/2 represents how big the aperture or hole in the lens gets. The lower the F/ number the bigger the hole, and the more light the lens lets pass through at any moment in time. The more light the lens lets in, the less amount of time you need to get your shot, meaning you can use faster shutter speeds to get the same result. It can make the difference between getting a nice shot or having too much hand shake or motion blur with a moving subject. Aperture also plays a big role in determining depth of field - how much stuff is in focus.

The sensor (ISO): The size (as well as type and quality) of the sensor effects the quality of the pictures as well as how noisy the pictures are. With all else being equal the larger the sensor the better the camera will perform as the ISO goes up. Higher ISO allows you to increase the shutter speed (to eliminate hand shake or motion blur), or increase the depth of field.

For even more detailed information: http://photo.net/learn/basic-photo-tips/aperture-shutterspeed-iso/ and http://www.cambridgeincolour.com/tutorials/camera-exposure.htm and http://www.kenrockwell.com/tech/shutteraperture.htm

Edit: mattdm does a fantastic job of explaining the 3 variables of exposure in his answer describing The Exposure Triangle!

Please let me know if anyone would like me to further elaborate on anything, happy shooting :)