Automatic (Green square)
As its name implies, automatic mode is purely automatic - not only does the camera decide all of the exposure parameters (and thus exposure) for you, under this mode you *can't* make any adjustments at all. Another attribute is that flash automatically pops up for Automatic; in other modes the flash will fire only when you manually pop it up. Obviously, this is for just using the camera to take photos and not learning photography, so most of you should be beyond this.

Advantages
Point and shoot without worrying about your settings
Less prone to screwing up if you don't have any idea what you're doing.
Disadvantages
Complete lack of control over exposure parameters
Flash automatically pops up in low-light situations whether you want it to or not.

Program Mode (P)
Program Mode is fairly similar to Automatic mode, with one key difference being that, although the camera automatically determines most settings, you have the *option* to override these settings. While the camera will determine an exposure and come up with a shutter speed and aperture (say 1/100s and f5.6) to take the shot, you can override this (scrolling the wheel dial in most cameras) with any other combination of shutter speed/aperture - say 1/50s f8 or 1/200s f4. This gives you the ability to go with automatic parameters as the default, but also quickly manually override, albeit not as efficiently as any of the Shutter or Aperture Priority modes.

Advantages
Point and shoot without worrying about your settings
Less prone to screwing up if you don't have any idea what you're doing.

Can override the automatically determined settings if desired
Disadvantages
Lack of immediate control over exposure parameters
Shutter Priority Mode (Tv)
Shutter Priority Mode allows you to manually define the shutter speed of the exposure, while the camera automatically determines exposure. Aperture and sometimes ISO (depending on the camera) will float to match that exposure. You can offset the automatically determined exposure via exposure compensation, usually by +/- 2 stops.

Advantages
Maintain a specific shutter speed when you don't care about aperture/depth of field
Useful for exposures with large depth of field anyway
Can guarantee fast shutter speed (and minimal blur level), at risk of underexposure
Disadvantages
Lack of precise control over aperture and depth of field
Doesn't guarantee fastest shutter speed possible
Underexposure if shutter speed is too fast for aperture, ISO limits
Aperture Priority Mode (Av)
Aperture Priority Mode allows you to manually define the aperture of the exposure, while the camera automatically determines exposure and lets shutter speed and sometimes ISO (depending on the camera) float to match that exposure. The exposure can be offset +/-2 stops via exposure compensation.

Advantages
Precise control over depth of field
When maxed out in conjunction with ISO, guarantees fastest shutter speed (and least blur) possible while maintaining correct exposure
Disadvantages
Possibility of dipping below blur threshold if scene too dark for max aperture, ISO
Manual mode (M)
In manual mode, every exposure parameter, including overall exposure itself, is manually defined. Thus the camera is unresponsive to the scene and captures a constant level of exposure unless you specifically change one of the three parameters.

Advantages
Complete control of three exposure parameters
Easiest way to shoot with constant exposure level (good for situations where lighting is constant - ex. many indoors in the same room)
Unaffected by metering errors (ex. dark clothing vs. bright clothing)

Disadvantages
Slow to adjust to non-constant lighting situations