Once you are level 7, you can start breeding your horses.

If you have an eligible stallion you must put it up for stud for it to be used for breeding.
If you have a mare that can breed, you will have to either request your own stallion for breeding or find a stallion available for stud to request.
Note: Mares and stallions must be 3 years old and be at level 10 or above to be eligible for breeding. Also, mares must be 50% or more happy but stallion can be any percentage of happiness.