Pokémon +

What are EVs?
EVs are 'Effort Values'; they are one of the things that change what attack, speed, defense etc of your Pokémon. IVs and DVs also change these, but you can't manually change them like EVs. Don't worry yourself with IVs and DVs; all you may want to know is that DVs are why a Pidgey is naturally faster than a Bulbasaur, and IVs are why one wild Growlithe has more defense than another, when at the same level. EVs are what your Pokémon get for battling other Pokémon with the kind of strength you're after.
For every 4 EVs of a stat, you get 1 added to that statistic.

What exactly do they do?
Well, we all know that Pidgeys are pretty fast Pokémon. They do, in fact, 'give off' Speed EVs. That means if you wanted to speed up your Geodude, you'd fight as many Pidgeys as possible, to collect their souls Speed EVs. Then, when the Geodude's experience brings it to the next level, you should notice it has a larger number for speed added than normal.

How do I 'EV train'?
First of all, recognise your Pokémon's weakness, or the strength it is lacking. Let's say you had a Mewtwo with barely any Special Attack. Without Special Attack, its 'Psychic' move would be a little wasted, wouldn't it?
So, we want to add Special Attack to Mewtwo.
What we want to do next is find out what Pokémon give off Special Attack EVs. Serebii.net has a nice list for us. Looking on that list, we can see that Gloom gives off 2 Special Attack EVs for each one you faint. Then, searching its Pokédex entry reveals that it is found on Routes 121 & 123 (Ruby & Sapphire). Gloom is actually a bad example as it is rare in that area, but we'll continue anyway.
Right. So you head off for Route 121. You get there, and encounter a wild Pokémon! However, it's not a Gloom. You can check to see if this Pokémon also gives you the EV you want, but it probably won't. In which case you choose to Run because you don't want their filthy no-Special-Attack-EV slime on your nice Mewtwo. Keep running from non-Glooms, and keep fainting Glooms.
That is how you EV train.

What are the limits of EV Training?
Obviously there will be limits or your Pokémon will be ultimate immortal beings. The maximum EVs total any Pokémon can have is 510, and you can only have a maximum of 255 for each attribute (like Defense). That means you could max out one stat, then strongly raise another, or equally distribute the 510 around the board of statistics.
There's probably no way you'd remember exactly how many EVs of a certain type a Pokémon has, so if you're maxing one out, just wait until one level no more of that statistic is received.

Is there any other way to raise EVs?
Yes. Stat-raising items such as Protein or Iron raise 10 EVs each for whatever stat they say. However, you can only use 10 of these items per stat total (meaning 100 EVs), and they are really expensive. The most consistent method of EV raising is actual training.

If you want lists of what Pokémon give off what EV, Serebii.net is your best place.

What are the best Pokémon for EV training?
It is dependant on the game you have. Here is a table: