It's actually quite similar to humans, except the method is a bit different :)
Mares (the female horses) go through cycles just like human women do, except in their cycle, they go into "heat" instead of menstration. When the mare is in heat, she is receptive to coupling (sex). The stallion can tell if a mare is in heat, and when she accepts it, he will mount her and do his business. The mare is always in control - if she's not ready or doesn't like the stud, she will kick him or bite him to let him know what she thinks.
The foal (the baby) is born approximately 1 year later, and will stay with his mother until he is weaned at about 1 year. I'm not a breeder, though, so I'm not exactly sure on these dates.
Hope it helped!