you should watch the video mentioned above, and look at the wiki page for it, but i'll tell you the basics. There are two modes, compare mode, and subtract mode. In compare mode, it will look at the input from the side, and the one from the back, and if the side signal is weaker, it will send out a signal, and if it is stronger, it won't. In subtract mode, it will subtract the side input from the back input and that will be the output. Also, if you hook certain blocks to the back of a comparator, it will give a signal based on the data from the block, for example, the more full a chest is, the larger the output, or jukeboxes will give a different signal depending on the disk that is in it. hope this helps

Watch sethblings comparator video, he explains how it works and what-not