When you defeat another pokemon, your pokemon gets some "Effort Values" is certain stats (HP, Attack etc). The stat you get depends on the pokemon you defeat.
The more EVs you get, the higher your stats become. The idea of EV training is to make your pokemon better in certain stats. If they have a good Attack you can EV train for Attack to make it great! Or you can boost flagging defenses and so on.
The link Swampert posted has plenty of information. You could try searching Google for more information, some of it gets complex though. :)