Also, while searching for pokémons for breedding you can use the following abilities, in addition to the other answers for your question:
- Magma Armor (ability): it's like Flame Body ability which half the steps required for hatching, if the pokémon that are using it's the first in the party;
- Trace (ability): this one, once in battle, will let you copy the target ability. So if you are looking for a specific ability, you will know what he have right when you enter in battle. But I want to point out that some abilities can't be copied, these are: stance change, trace, multitype, imposter, flower gift, illusion.
- Cute Charm (ability): with this ability in you first pokémon in party, your chance of encontering pokémon of OPOSITE gender in the wild will be 66.7%! e.g. if you have a male Minccino with cute charm in your first slot, when you enconter a pokémon in the wild, you will have 66.7% of encontering a wild female pokémon. This is great for when you are in search for a female starter in friend safari (they have very low chance of beeing female).
- Skill Swap (attack): same as Trace, but it's an attack.
- Synchronize (ability): as stated in other posts, give you 50% chance of encontering a pokémon with the same nature of the first pokémon in the party
- Power items: these items make sure that 1 IV will be based down to the baby by the parent that is holding the power item. Great when starting and you don't have the Destiny Knot yet.
- Destiny Knot (item): Will make sure that 5IV will be passed down by the EITHER parents. Without this item, only 3IV is garanted to be passed down.
These are all items, abilities, attacks, that I know so far. Also, here are some tips for finding pokémon:
- Try to get as many 31IVs Ditto you can. Since every breedable pokémon can breed with Ditto, a good way to start is getting these Dittos. Use pokémon with sinchronize to help with getting Dittos with the nature that you need. Also, named the Dittos with the nature that they have, to easy find out the one you are looking for while in pc.
- Another tip is to use a pokémon with synchronize AND Skill Swap, if you are looking for a right to breed pokémon with desirable nature and skill. For this, you will need to breed a MALE Reuniclus Lv.45 w/ Skill Swap and a FEMALE Ralts (preferable with Synchronize to increased chances of passing down ability) to get a Ralts with Syncronize and Skill Swap.
- One more tip, that I found out to be the best if you already have the Ditto with the desirable nature for breeding, is to make your first pokémon to be a Male Cute Charm, and the second be a Trace one. With this, you first make sure to knock out your first pokémon (cute charm still activates even if the pokémon is fainted), then when you search for an pokémon with a desired ability, you'll already have a pokémon to increase chances of getting a female pokémon (for those with low female chances) and the second, that will now start the battle, can make you sure that you know the ability of the target pokémon before deciding whatever you'll catch it or not.
- Last one is to have a Parasect. He can learn both False Swipe (to weaken wild pokémon to 1HP - but doesn't work with ghosts) and Spore (100% chance of sleep - but doesn't work with grass pokémon), so lead with one of the pokémons above and switch to parasect to weaken and capture in the wild.
I hope it helps while catching your pokémon!
EDIT: Remembered more 2 items for breeding:
- Everstone (item): This one make sure that a nature of the parent holding it will be passed down to the baby. This is great combined with Destiny Knot after you have the desirable parents to pass down IVs, nature, egg moves
- Oval Charm (item): This one increases the chances of receiving an egg at Day Care. In Pokémon X and Y, the player can receive the Oval Charm from Professor Sycamore after seeing all Pokémon in each section of the Kalos Pokédex, with the exceptions of Articuno, Zapdos, Moltres, Diancie, Hoopa, and Volcanion.
In Pokémon Omega Ruby and Alpha Sapphire, the player can receive the Oval Charm from Professor Birch after seeing all Pokémon in the Hoenn Pokédex, with the exception of Jirachi.
Some information was got from bulbapedia, others are experience.