(for more advanced information please see Combining Monsters 101 )
All combining of monsters is done at the academy.
Primary Monster: More traits will be inherited from this monster.
Secondary Monster: Fewer traits will be inherited from this monster.
If you want to keep a certain species then a monster of that species should be the primary monster. If the same species is chosen for both the primary and secondary monster then there is an almost 100% chance that the resulting monster will be of that species, though random alternatives have been known to happen.
Generally speaking you should breed monsters which are retired so that the resulting monster will be as strong as possible.
You can produce a new monster by combining two of your older monsters. Both need to be frozen before a combination can occur. The new monster will carry the traits of both parent monsters, including the distribution of trained stats and techniques.
Your monster must be the primary monster and the neighbors monster must be the secondary monster. You only need one frozen monster of your own. Your monster will disappear, but the neighbors monster will not. (And therefore you do not need their permission to breed. It just happens.)
When combining monsters you can use combination items to increase their stats or increase the probability of a particular monster being created. Look at combination items more information.