Breeding in CryptoKitties is designed to act like breeding in real life with respect to the passing down of traits / genes. There is a Dame (the female kitty which will birth a new kitty) and the Sire (male kitty). Any kitty can play the role of Dame/Sire, but you cannot perform both activities at the same time. Both the Dame and Sire's genes are mixed together to produce a new kitty.
Just like in nature, some traits are recessive, and some skip generations. When breeding, you choose one of your kitties to be the Dame. You must also choose another one of your eligible kitties to be the Sire, or rent one from another user, via the CryptoKitties marketplace. Breeding is performed via the CryptoKitties website