Basically, EV training in G3-5 gives you an edge by concentrating on at least 2 or 3 stats on a given pokemon. A pokemon EV trained in attack will have more of a punch in offensive power than a non EV-trained pokemon. A pokemon EV trained in defense would theoretically take lesser damage, in speed the pokemon would be faster, etc. No matter the trainer, you want to win. Almost everything in-game has an impact on the metagame.
In addition to having concentrating stats, having the right nature can add more oomph and having the wrong nature can nullify your efforts. When you EV train, it's not uncommon to EV train the pokemon's highest stats. For example, Raichu's highest stat is speed. Ideally you'd want a Raichu with a nature that can boost its speed. The natures that boost speed are timid, hasty, jolly, and naive. The second thing you want to look at are the pokemon's learnable movesets along with its second highest stat. Raichu has a tie in both physical and special attacks as its second highest stat and can learn plenty of moves that fit either stat. If you want a Raichu that specializes in both speed and attack, you don't want a Raichu with either timid (lowers attack) or brave (lowers speed). Likewise with a Raichu concentrated in special attack, you'd want to avoid the natures adamant, impish, careful, and jolly. There are also five natures that are neutral and have no effect if you want to EV train which can be good (if you're not too picky about certain natures like myself) and bad (if you're deadly serious on one specific nature).
With the advent of the power items introduced in G4, EV training has become a lot easier and a lot faster if you play your cards right. If you can manage to get Pokerus and a power item attached to the pokemon you want to EV train and say defeat a Crobat which gives 3 EVs, the power item would increase that by four to about 7, and the condition Pokerus would double that to 14 EVs in a single battle. I think I might be wrong in the math, but that's the basic idea. The vitamins introduced since G1 are also very beneficial to EV training because they can be taken at least 10 times, giving 10 EVs each time, 100 EVs possible. In Gen 3 with the new berry concept, there are 6 berries can decrease the EVs from the stats if you accidentally increased the wrong stat and reverses what the vitamins give.
There's also something called IVs, short for individual values, that a pokemon has to their stats but I'm not that knowledgeable in that area. I hope the explanations I gave aren't too wordy. xD