I'll start with the more common and well known modes:
The most commonly used mode on any camera. In this mode, the camera selects what it thinks are the best settings based on the conditions in which you are shooting. Auto mode can be great, however, the camera does not always see what you see.
Portrait mode: In this mode the camera will automatically choose a larger aperture (smaller F-stop number) to make your subject in focus but the background a bit blurry. You aren't going to get the major blurred background like you will with a DSLR (that's another tutorial) however your background will be softened allowing your subject to be the main focus of the picture. Your camera will choose the focal point based on the closest subject to you, therefore, this mode is great if you are shooting one subject (or possibly more than one subject if they are at equal distance from you).
Action mode: This mode may also be known as fast shutter speed or sports mode. In this mode your camera automatically chooses a fast shutter speed to help freeze action. This allows you to keep a moving subject in focus without motion blur. To best use this mode, pan your camera along with the subject while shooting because your camera is continually focusing on that subject. You can also attempt to pre focus your camera on the spot where the subject will be when you want to photograph it but that's a bit hard.
Slow shutter speed mode: Pretty much the opposite of action mode. This mode uses a slow shutter speed to allow your moving subject to be blurred. This is great when you want to get a picture of flowing rapids or waterfalls or when you simply want to take a cool blurred action shot.
Landscape mode: In this mode your camera automatically chooses a small aperture (large F-stop number) to make sure as much of your scene is in focus as possible. This is ideal for capturing subjects that are at different distances from you. You may need to be careful of camera shake when shooting in this mode as your camera may try to compensate for the small aperture by selecting a slower shutter speed.
Child mode: My camera does not have this mode so I don't know much about it. From what I can find, it is similar to a combo of portrait and action mode. It allows you to get a great shot of a moving child while keeping colors vibrant, skin tones natural, and the background slightly blurred.
Macro mode: This mode may also be known as close-up mode and is one of my favorite modes. You won't always find this mode on the dial selector on your camera. You may have to look elsewhere. On my camera it's below the flash selector. This mode allows you to take super close-up pictures of flowers, insects, or other small objects. Every camera is different when it comes to the distance ranges it will shoot in macro mode. I could write a whole post on this mode, and just might! One thing to remember about the macro mode is that you don't want to use a flash or you will over expose your subject. You may also have to worry about camera shake depending on your focal distances.
This rounds out the easiest and most common modes on a point-and-shoot camera. Choosing Your Camera Mode-Part 2 will focus on those strange and scary modes on your camera (Tv, M, S, Av, etc).