Ok, I'm going to take a stab at the basics of Breed ID's.
Breed ID is basically a number representing the stat allocation. All wild pets
* have a set of breed ID's that they can have, based on their specific type. The breed ID's are divided into two sets, that are exactly the same, EXCEPT that one set indicates male, and the other indicates female.
Male ID's are 3-12
Female ID's are 13-22
Some breed ID's are popular for their focus in one stat in particular. Sometimes folks will just refer to the male ID#, as opposed to the male and female #'s. There is no difference, stats wise. For example the breed ID 4 is known for having the highest attack at lvl 25. Breed ID 5 is the highest speed. These Id's are shown on the "Battle" tab for any pet listed on this site. You can see a relative comparison, shown by this graph on the right side of the Battle tab:
This is the graphical representation of the Breed ID 12/22
You also mentioned quality (as in poor, common, uncommon and rare). Think of the quality as a multiplier on the stats of the pet. The higher the quality of the pet, the better multiplier you get.
The stat allocation for the pet is determined by the Breed ID and the quality acts as a multiplier.
*Wild pets will (to my knowledge) have the same stats if they have the same breed ID and quality. However, pets obtained by other means are not consistent. For example, Lil' Ragnaros
has a higher attack than wild pets who share his Breed ID 4.