Digital SLR cameas for a long time have a 3:2 aspect ratio in their images. That means the width compares to the height as 3 to 2. That does not surprise, because we were used to that aspect: all printed pictures had formats like 6″ x 4″ or 15×10 cm. That is why this is still referred to as the picture-format. Popular resolutions (number of pixels in your image) are 3456×2304 (Canon) or 3872×2592 (Nikon).
If you want to print your pictures, this should be you favourite choice.

Compact cameras in contrast used a different ratio: 4:3
This one also has a long history: the first movie equipment, early in the 20^th century had this format. It also got an enormous boost when the PC was invented: earlier and current computer screens have resolutions like 640×480 or 1024×768 and the very nice 1400×1050 flatscreens you can buy now. That’s why this ratio is called (S)VGA.
If you mostly use your pictures on a computer screen, this one is the best to use. By the way, photolabs sometimes offer you the option of 15×11.5 cm prints: use that one for the usual pictures from a compact cam.

A pretty new format is 16:9 which is introduced by modern television technology, often called HDTV.
Several new digicams can be set to this ratio giving resolutions like 1280×720 or 1920×1080. Very nice and very wide indeed.
Now more and more displays have dual functions (computer + TV) this might be a very common ratio for the future! So, if you display your digicam images mostly on such a screen or HDTV, this is the right setting to use.

The very latest BenQ cameras even offer you both 3:2 and 16:9 options for your pictures…