To answer your question in the very basics:
Each pokemon in the wild gives a certain amount of effort values every time you kill it. These effort values, depending on the pokemon, contribute to your pokemon's devlopment in hp,attack,special attack,defense,special defense, or speed. The maximum effort value total for each pokemon is 510. After 510 effort values (a bunch of pokemon have been killed) have been distributed to a specific pokemon in your party, effort values stop accumulating. The art of EV training is to train your pokemon on certain species to get less than or equal to (a maximum) of 252 (maximum EV allowed per stat) EV in a stat that you want to strengthen on a pokemon.
That's probably confusing, i hope not, but that's just really really the basic goal.
You breed pokemon to breed something called Individual values (which further enhances the power of each stat, maximum is 31). IV's are the basis for the phrase "no two pokemon are alike". The max IV for each stat is 31, although it is extremely rare (HACKS) to have 31 in each stat. IVs are passed down by parent pokemon (three IVs per parent) into the offspring.
ex. Ditto + Eevee
Ditto has a/b/c/d/e/f (each letter is an IV, and is in the spot of hp/attack/sp atk/defense/sp defense/speed)
Eevee has u/v/w/x/y/z
The offspring eevee (in the 4th-gen) gets three IV from each parent. (In the 3rd generation there's a glitch that lets the same stat get inherited if both parents give the same stat by chance, so the pokemon would only inherit 5)
new eevee has a/v/c/d/x/y
You also breed for a correct nature. (Each nature with the exception of ....4 i believe promote one stat and demotes another.)
Those are really basics. I remember I was so confused when I first started EV training and "competitive breeding/raising"
Go to gamefaqs and serebii. There are some great guides on gamefaqs that can answer all of your questions. It took me about 2 weeks to get the full hang of it. But I didn't spend my whole time trying to get it. It comes into place.
Seriously, go to gamefaqs!!!