EVs, or "Effort Values", are external values important to the overall training
of a Pokemon. They are gained, along with Experience Points (EXP), from any
Pokemon you battle, may it be in the wild or from other trainers (though, as a
side-note, Pokemon from the Battle Frontier do not give EVs, as they do not
give EXP as well).
Each Pokemon has a unique set of EVs they give out, 1-3 EVs for one, or
sometimes two, stats. For every four EVs a Pokemon obtains, that's one point
added to the stat with four EVs. For example, if you fight four Sentret (which
give 1 Attack EV each), you gain 4 Attack EVs, which means +1 in the Attack
Sounds neat, right? But, of course, there are limits. A Pokemon can only gain a
total of 510 EVs, and 255 EVs per stat. That means the maximum stat bonus EVs
can give per stat is 63, should you give 252 EVs to one stat. 252 is the
accepted bonus, as it is the highest number less than 255 divisible by 4. With
that said, 252/252/6 is pretty much the most basic EV spread for a Pokemon,
adding 63 points to two stats and 1 point to one stat.
So yeah, it's very important. An un-EV trained Pokemon almost always falls to a
properly-EV trained Pokemon of the same level. Add in IVs and Natures, and
you've used your Pokemon to its fullest battling potential.