Think of IVs as a DNA code.. a genetic code.
Each Pokemon get a set of IVs for each of their stats which can go from 0 - 31 for each stat. (0 being the worst IV to 31 being a perfect IV.) Those numbers can make a huge difference because it can be the difference from you surviving an attack, from you killing the Pokemon on your turn, to your Pokemon going first. This is why even if you have two Ninetales, one ninetales will go first before the other one. In competitive play, it's important to have have perfect IVs, which is 5 31 stats for most Pokemon. (Like you wouldn't want a Pokemon you're going to only make a magic attacker have good attack stats.. so that's why perfect is considered at least 5.. unless you have a mix attacker that relies on both attacking and magic attacking.)
In Pokemon X and Y, it's relatively easy to breed out perfect IV Pokemon. If you start out with two random Pokemon you found with bad IVs, you can make good breeding parents in about 30 minutes - 2 hours depending on what you're seeking. (If you're seeking a hidden ability.. egg moves, etc.) After you have two perfect parents in the daycare, you can breed out "Perfect" Pokemon all the time in 5-30 minutes.
To pass on the IVs, you need a Destiny Knot. A Destiny Knot provides a CHANCE to breed 5 random IVs down from each parent. Take not you don't always breed down 5 IVs, because you can put in two 5 IV parents and come out with crappy IVs, but it works at a pretty decent chance.
http://destinyknot.tk/ (Bookmark this. This is a way to tell what your chance of getting certain IVs is by putting certain Pokemon in the daycare.)
And I know you're wondering this:
How do I know the IVs of my Pokemon? Well I'm glad you asked!
The final town that you're invited to, I forget the name of it, but you go to it after you beat the Pokemon league from the train. There's a guy in the Pokemon Center that will tell you the potential of your Pokemon. If he tells you a certain stat "Cannot be beat!" then that means it has a perfect IV in the stat he tells you.
NOW for EV training.
EV training is the mode called "Super training" where you make your Pokemon beat up balloons with a soccer ball. Completing a full super training session takes about 30 minutes I believe. You can max 255 (or 256) points into two stats, and put a small amount into a 3rd one, or you can spread it around. There's different modes that will give you different amounts of points. I personally just stick to the easy ones, because I find it faster (Unless I have a double up bag.)
You can also EV train by finding horde battles, but I'm too lazy to figure all of that out.
You can watch this if you're interested:
http://www.youtube.com/watch?v=kR6uzfwcNjY
It's much faster, but I'm lazy O_O
Also you can reset all your EV trained stats, so you don't have to make them perfect. If you punch the bag long enough on the super training menu, you'll get a reset bag.
IVs you can't change.