Basic Camera Modes

People are often a little bit confused on what some of the things on the mode dial on their camera does, and which they should use, and when. What symbols and abbreviations are used varies a little, but the basic concept is the same, in general. The standard for me is the set that Nikon uses on its SLRs, but you should be able to figure out what's what given the following information.

Green = Take pictures mode
P = Take pictures mode, but stop popping up the flash all the time mode
S (Tv) = Take pictures at the shutter speed I specify mode
A (Av)= Take pictures at the aperture I specify mode
M = I'll figure own damned exposure, thank you very much mode

Green - The first mode is usually a symbol of some kind instead of text, and it's usually green, so that's what I'm calling it. The "Playback" mode, is also usually green, so don't get the two confused, or you will be very lost.
Green mode is generally what you want your camera on if you hand it to someone else, as it lets the camera make its best guess at what the picture should look like, and often has a simpler interface than the other modes. A good place to start, if nothing else.

P - Stands for "Programmed Exposure". On a Nikon SLR, this is mostly the same as Green, but gives you control of whether or not the flash pops up when the camera needs it.
I use this as my default mode, simply because I don't like having the flash pop up all the time, and I never take pictures in decent light.

S (Tv) - Stands for "Shutter Priority" mode. On Jes's Canon Powershot A540 (an excellent camera I could recommend to nearly anyone), this is Tv, but I don't know what that's supposed to stand for.
In any case, what this mode allows you to do is set the shutter speed, and the camera will try and get the correct exposure by varying the size of the aperture. Shorter shutter speeds (1/125 is shorter than 1/30) let in less light, but also combat shake and subject movement.

A (Av) - Stands for "Aperture Priority" mode. Again, note about other cameras' abbreviations.
This mode allows you to select the aperture and let the camera worry about changing the shutter speed to match.
Larger aperture numbers (f/32) indicate a smaller hole through which light can hit your film/sensor. This inverse property is due to the fact that the f-number (ex: f/32, f/8.0) is a ratio of the focal length (zoom) of your lens compared to the size of the aperture. So as the aperture gets bigger, the ratio becomes smaller because it approaches 1:1.
What does this mean to you?
Large apertures (f/2.8 is pretty big) let in a lot of light, which is good for low-light photography, but give you a very small depth of field (the area that's in focus).
Small apertures (f/16 is small, but f/32 is smaller) give you a larger depth of field, which means you don't have to be quite as worried about getting a super-accurate focus. On the other hand, smaller apertures let in less light, so you are more likely to lose a picture due to shake when you use smaller apertures.
Zooming in generally forces you to go to a smaller aperture, so if there's not much light, zoom all the way out*.

M - "Manual exposure" mode. You control the shutter speed and aperture, which means the camera will not help you much at all in terms of getting the picture to come out at the right brightness. Depending on the camera, it might mess with the ISO for you, but you can turn that off. If you do, it's entirely up to you to get the picture right. You have to balance shutter speed and aperture, along with ISO, to expose the picture. Your camera will usually still help you figure out white balance and focus, however.

*For those of you that have $1,000 to drop on constant-aperture lenses where this wouldn't be the case, I expect you to know what I mean, and to know that it doesn't apply to you.
I'm also envious.