There are a lot of resources out there for EV training, so I don't want to spend too much time on this subject.
I'll summarize it.
EVs are Effort Values. Basically, every single Pokemon is worth amount of EV points in a certain stat.
For example
Psyduck is worth 1 Special attack EV. But Golduck is worth 2 Special Attack EVs.
To find out what any Pokemon is worth, check out this link:
Each Pokemon is able of earning 510 EVs in total. But, a max of 255 EVs per stat.
4 EV's = 1 Stat point extra at level 100.
Because of this, it's always advised to put only 252 max EVs, otherwise the last 3 EV's will be unused and wasted, give a max of 252 EV in two stats, and 6 in a third to optimize EV training.
Now, you may think EV training is a bit tedious, given a Psyduck will only earn you 1 in special attack. Well, there are items that speed this process up, along with a strange Pokemon Virus called Pokerus.
Macho Brace:
The Macho Brace is for EV training, Basically, any Pokemon holding this item, will gain twice the amount of EVs as normal.
-1 Psyduck = 1 x 2 = 2 EV
Pokerus:
Pokerus is a strange virus that doubles a Pokemon's earned EVs, it's really rare, and you may just catch it randomly when you battle a wild Pokemon that has it. But it's really rare, up there with the shiny Pokemon. So its advisable to hunt down someone who has Pokerus on their game, and get them to spread it to your Pokemon.
It's always good to keep a couple Pokemon with the virus in your PC and not to touch them, only use them to re-spread the virus across more Pokemon, should the virus disappear from them, which it does over time.
Though, if a Pokemon had Pokerus, it never goes away completely, they may lose their infectious ability, but they don't lose Pokerus' ability to Double the EV's a Pokemon earns.
-Battle 1 Psyduck with Pokerus = 1 x 2 = 2 EVs
-Battle 1 Psyduck with Pokerus while holding Macho Brace = 1 x 2 x 2 = 4 EVs
Power Items:
Power Items can be purchased at the Battle Subway (or Battle Frontier). They're really useful items for EV training. There are 6 of them, one for each stat:
HP - Power Weight
Attack - Power Bracer
Defense - Power Belt
Sp.Attack - Power Lens
Sp.Defense - Power Band
Speed - Power Anklet
Each of these items will add 4 extra EVs on top of the EV(s) it earns in the stat the item is meant for.
-Battle 1 Psyduck while holding Power Lens = 1 + 4 = 5 EVs
-Battle 1 Psyduck while holding Power Lens and with Pokerus = (1 + 4) x 2 = 10 EVs
-Battle 1 Golduck while holding Power Lens and with Pokerus = (2 + 4) x 2 = 12 EVs
That is all there is to EV training really, the only finer details I can think of, is that if a Pokemon holds EXP share, then yes, they too gain EVs, but they only get the normal amount, unless they have Pokerus, in which case, they get double. And EVs don't get shared between those who were in the battle, they all get the EV as if they battled it by themselves.
EV training Hot Spots:
HP - Ducklett (Driftveil Drawbridge, shadows) = 1 EV
Stunfisk (Iccirus City, Surf) = 2 EVs (I'd advice using the stunfisk)
Att - Lilipup and Patrat (Route 1) = 1 EV
Def - Sewaddle and Venipede (Pinwheel Forest) = 1 EV
Swadloon and Whirlipede (Pinwheel Forest) = 2 EVs
S.Att - Litwick and Elgyem (Celestial Tower) = 1 EV
S.Def - Frillish (P2 Lab, Surf) = 1 EV
Speed - Basculin (Water just off Route 1, Surf) = 2 EVs
I hope this helps you with your EV training. Just remember to count the EV's properly.
If you need to check to see if they're done properly, go to Opalucid City, in the house behind the Pokemon Center should be a girl who tells you if they're done.
Not done = Hasn't worked hard enough yet
Done = Oh, your <x> has worked really hard!
Stay tuned for more on IV breeding, because IV breeding makes full use of a Pokemon's potential, and without it, I honestly don't see the point in EV training.