I know that in the full auto (green) mode, you cannot use Exposure compensation. I believe the same may be true for the 'icon modes'. This is one of the main reasons that I recommend people don't use those modes...ever. So if you see that EC is greyed out...I'd guess that is the reason.
If you aren't ready for the priority modes (Av & Tv), or for manual mode, then I'd suggest using P (programmable auto). This mode will allow you to use EC as well as several other options that aren't available in the fully auto mode.
Exposure compensation can be very important when shooting any any of the auto/priority modes. It's a bit of a long lesson, but the gist of it is that your camera is designed to get exposures wrong in specific situations and in order to actually get properly exposed photos, you must use exposure compensation (or manual mode).
Note that EC doesn't do anything special or magical, it simply changes your exposure settings (shutter speed, aperture or ISO) in relation to the metered value.
AEB (auto exposure bracketing) is a mode where the camera will take three successive exposures at different exposure values. You can set how far in either direction you want to vary the exposure. For example, you can set it so that the three photos will be taken at zero (no EC), +1 and -1. So you end up with three photos with different exposures. You can then (later) choose the best one. Or you can use software to combine the various exposures into one photos that has a higher dynamic range (HDR) than a single photo could have.
So in a nut shell, EC is something that you should use frequently, because it may need to be different for any given shot you take. AEB is typical something that people only use when they are taking their time (maybe landscape photography) and/or when in tricky lighting situations.