I'm also new to the competitive scene, and trying to get into it. I've found that watching battles on Youtube helps me to see strategies (whether it's VGC style of Doubles, or 6v6), and a couple people even make team building videos (Shadypenguinn does this) which goes over the thought process of a particular team.
If you plan on playing by the Tier system, people play by
Smogon's rules, so you'd definitely want to check them out to see what Tier your pokemon falls under (it looks confusing and complicated at first, but it gets easier to understand).
Again, there are also a few Youtube videos which go over that as well.
Like Earth Wolf said, competitive is highly based off of IV points (Individual Value) and EV points (Effort Value).
The higher the IV is in one of the six stats (with a max of 31), the better/higher the stat will get. When breeding, it's rare to get a 6-IV pokemon (which means perfect IVs, 31, in each stat), but a lot of people aim for as many perfect IVs as they can get, or a spread which will give them the Hidden Power they want (as the Type of Hidden Power is based off of the IV spread of a pokemon).
EVs are what you'd use Super Training for, and increase the number of a stat. The max EVs a pokemon can have in a single stat is 252, and the max EVs total (spread out amongst the six stats) a pokemon can have is 510.
Nature is also important in competitive, depending on the role the pokemon is supposed to fulfill. Bulbapedia has a
list of natures, which can be helpful until you have them all memorized (or not...I know what a few do off the top of my head, but I haven't memorized every single one).
Pokemon are also bred for Egg Moves, which comes from breeding pokemon with a particular move with the desired pokemon, so the one which hatches from the egg will have that move.
Serebii's pokedex lists all egg moves a pokemon can learn (as well as TMs and HMs), and is a great resource.
A pokemon's Ability is also important to consider. Some pokemon have 3 abilities (two normal, one hidden). Depending on what you want its role to be and what you want out of your team will depend on what ability you choose to breed for.
Finally in the breeding category, are the tools you'll need to get the proper nature and IVs. You'll want the pokemon whose nature you want to hold an Everstone, and the other pokemon to hold a Destiny Knot (as this guarantees passing on 3 perfect IVs to the child).
To check your IVs, there is a guy found in the Pokemon Center in the Battle Resort (or, the Pokemon Center in Kiloude City if you're using X/Y). In ORAS, he's standing in front of the map of the region; in X/Y, he's tucked away in the left corner right before you would continue to change clothes.
Depending on what he says will tell you which stats have perfect IVs. Again, Bulbapedia has a
list of what he will say and the range of IVs that correlates to.
From what I understand, there are roles in competitive battling, which can be broken down into:
Physical Sweeper (typically max Atk and Speed stats, usually Adamant or Jolly nature)
Special Sweeper (typically max Sp.Atk and Speed stats, usually Timid or Modest nature)
Mixed Sweeper (often a nature sacrificing one of the defense stats, with EVs usually in both Atk and Sp.Atk)
Physical Wall (typically max Def stat...perhaps HP, as well)
Special Wall (typically max Sp.Def...perhaps HP max, as well)
There are other roles (such as stalling and setup), but most of the 'mons will usually be bred to fall into one of the above categories. Important moves to consider are Rapid Spin and Defog (for 6v6), as those moves get rid of entry hazards like Stealth Rocks and Sticky Web.
Hold items are also important, and most of the ones people use are won from the Battle Maison. Things like Choice Specs, Choice Scarf, Choice Band, and Life Orb are often used. The Choice items will lock you into a single move, which means you might have to switch to another pokemon if you run into a poor matchup.
An item like Assault Vest will help bulk up a Special Wall, or just make a pokemon harder to take down.
All of this, I learned by either looking it up on Smogon, or watching competitive battlers on Youtube (and some casual battlers), as well as the Pokemon GBA competition currently in progress on Youtube, as well.
If I got something wrong, someone else can feel free to correct me (as I'm still learning, myself). ^^