Unlike the plethora of information sites detailing the mechanics of breeding, I will present the new mechanics in a "how to" style as a means to, hopefully, help those new to the game, and those unclear of how it works understand it better. First you will need a lot of patience, unless you're one of those who managed to find a perfect breeding Ditto. It's also good to have one or two boxed in the PC empty so you can fill them with eggs for a less tedious method of hatching.
You will need a few things in the game, first you need access to the stat rater in Kiloude. You will have to wait until you are past the E4, Kiloude is also where the power items are. You will need a destiny knot as well, this is your most valuable item if you are going to be a breeder so make sure you don't lose it. The other items are the power items, their names all start with the word "power" and you can get them all in Kiloude using Battle Points earned in the Battle Maison or the Battle Institute. The exchange desk on the left is where you can trade the points for the items.
A single Ditto is all you may need, though having a few with some maxed IVs can help a lot. I used this method to get a Garchomp with all six IVs perfect, using only a Ditto with no maxed IVs and about 100 eggs. So a "perfect" Ditto is not required at all. If you want to breed egg moves, and your first one is a female, you can do that now. If you don't have a female then just breed with the Ditto to get one, the number of eggs will depend on the chances of getting a female. Those without a gender cannot get egg moves so don't bother looking for that information in such a case.
Egg moves are the same and there is a ton of information on how to breed them into the pokemon. But make sure that both parents you choose to continue on with have the egg moves you want your perfect Pokemon to have for the next portion of breeding. Now you are ready to breed for the IVs in generation VI. First, check the IVs for the parents, make note of those the rater says "can't be beat." If both parents have the same ones then you will need something to breed the female with for adding more, otherwise you will need a lot of luck and patience. If neither have any maxed IVs then breed just those parents with no items held, do not worry about the nature yet.
Breeding whil the parents hold no items will ensure that a few stats will be completely randomized, giving you a chance to get one or more maxed IVs in the offspring. Having two parents with different maxed IVs is your best outcome, even if it's only one each.
Now that you have a base pair, you can begin the controled breeding process to improve each new generation. If you have a parent, or Ditto, with three maxed IVs or more, then you will want to skip this next part. If all you have are one or two maxed for each parent then attach the power item corresponding to the stat you want to ensure gets passed to the next generation. For example the power anklet will ensure speed is passed to the offspring, it's the stat that grows when held. At this point, about thirty eggs will guarantee you get one that you are looking for, possibly even a better pair. Again, try to get a pair with different IVs, even if that means one won't be better than the parents.
Now you are ready to use the destiny knot in your breeding, so I will attempt to explain how the destiny knot works now. The destiny knot will ensure that five stats are inherited from the parents, however, which stats are inherited and from which parent are random. This means that HP, attack, defense, special defense, and speed may be inherited, maybe three will be from the mother and two from the father. Combining the destiny knot with a power item does not change that much, but does offer some control over who the stat is inherited from. For instance, if a parent is holding the power anklet and one of the stats inherited by the destiny knot is speed, the speed will be inherited from the parent holding the power anklet.
This also means you will need to change the parents a lot, often going through your boxes to find a compatible male parent with a stat you want to add to the maxed ones. This is when things can get frustrating, the key is patience, you will need at least thirty eggs per generation untill you get one, or two, that are exactly what you expect. If at any point you end up with two parents that each have three or more maxed IVs that cover all six stats you can move on to the next step.
Now you should have two parents which, if combined, cover all the stats you want, exactly how you want them. If one has the nature you want as well then you're almost done. Only one parent will need the destiny knot, and it does not matter which parent you give it to. If one of the parents has a nature you want for your final result then they have to be given an everstone now. If you have no interest in either nature then do not use the everstone at all, just the destiny knot. The next generations you breed should present a breeding pair that is at least one better than the previous parents, again make sure that the stats for the pair are different and cover all the ones you need maxed. Keep an eye out for one with all six stats exactly what you want, in a batch of thirty eggs you have a good chance of that happening, in a batch of sixty eggs you will have a perfect breeding pair and a great chance of one being all six stats exactly what you want.
If you are still trying to get the perfect stats and nature then just keep repeating the last step in the process until you do, it should not take much longer now. I should mention hidden abilities now, though you may already know about this. The female of the parent, if they have a gender, has the best chances of passing a hidden ability on. The male has almost no chance of passing it on, even with a Ditto for the other parent. That means it's best to keep only the females with the hidden ability through the entire process if that's the ability you are interested in, or keep one on the side to breed it back into the final generations if you want it to move along faster.
I hope this helps some players breed more competitive play pokemon for interesting battles. Happy breeding, and I hope to see you all on the rating lists.