Breeding is the process by which the player compels two puppies to mate, producing an offspring that combines its parents' traits via simple Mendelian inheritance. This command is issued with an upper-case "M" and is an important, vital part of obtaining the Sasquatch Hide Armor. It is not, under any circumstances, to be confused with the lower-case "m" command, which causes your dwarf to mate with the nearest creature. Mating with a puppy will always cause God to unleash the smite attack.
The point of breeding is to emphasize desired traits and to eliminate undesired traits. Since a perfect puppy with high scores in all traits is impossible-- the game assigns points to each trait depending on its parents' traits, but almost never in excess of a base amount of 10 positive points-- your modus operandi should be to breed together parents who are both strong in the traits you want and weak in the traits you don't.
Since you're unlikely to find such a combination right off the bat, or indeed in the first two or three generations, you'll have to breed dogs as placeholders, to get to the parent couple you ultimately want. It's important to level up one of the parents for each couple, because its increased stats will override the ten point rule and give you a precious two points of breathing room.