EVs are points you gain for each pokemon you KO. Every 4 EV Points adds 1 to that stat. There is EVs for every stat. You get a maximum of 510 EV points, but the max for each stat is 255. It is best to get 252, that way you do not waste any points.
So, let's say that you EV Train on Bibarels. Bibarels are worth 2 EV points for Atk. So, with 2 Bibarel KOs, you get 1 Atk Point. That means that by KO'ing 176 Bibarel, you will have the maximum EV points in Atk. 252/4=63 Atk points tacked on to your Pokemon. So, let's say for sake of the discussion, your Flareon reaches level 100 and you EV trained with Bibarels and Staravias only. Instead of just 359 being the max for your Atk, it becomes 422. The 229 Speed becomes 282. That is the difference between those that EV train, and those that kill random pokemon.
*takes a breath* Any questions?