When a request has been approved for a stallion and the mare is eligible to be bred, she will have a little truck icon appear in Mare Services. Clicking this will ship her to the stallion so that she can be bred. If you do not ship your mare, she cannot be bred! Note: This 'shipping' does not actually remove the mare from your care/barn.
Pregnancy risk will be initially calculated based off of the mare's energy and condition level at the time of breeding. It can go up or down based on those levels throughout gestation. Having a high pregnancy risk can cause stillborns, mare injury, or even mare death, so it's best to have your broodmares doing Groundwork as their exercise for the month.
For stallion owners, there is a link on Stud Services that says 'Breeding Center'. Going there will show a list of your stallions along with a list of any/all mares currently shipped to them waiting to be covered. It will also show the stallions' current energy level. From that page, you can click the white heart to cover whichever mares you choose. Once covered, the mare will disappear from the list and will also show the covered (half-heart) icon on her owner's Mare Services page.
If a mare does not get in foal, she will have to be shipped again once she's eligible to make another breeding attempt. This will send another PM to the stud owner and the process can be repeated just like it was the first time. Covering a mare on a rebreed will use a stallion's energy, so keep that in mind. It does not, however, take an additional breeding slot (for the mare owner) or charge an additional stud fee.
When a stallion covers a mare, it will use energy! This is very, very important! A stallion can only cover as many mares in a month as his energy allows. The amount of energy used per breeding is based upon his current condition level. A stallion who is in poor condition will only be able to cover one mare per month, which will also naturally limit the amount of mares he can cover in a year.
A stallion in optimum condition will be able to cover up to 3-5 mares per month, meaning he will be able to have more foals born within the first months of the year. It is not all that difficult to keep your stallions in decent condition. Unless you are skipping training or abandoning your account for long periods of time, most stallions will always be able to cover at least 2 mares a month.