First find the stud you would like to breed the mare to by using the icon option under the icon and selecting stallions and checking the stud box. You can then select the stallion and the short description that comes up will have a request stud link to click. A drop down menu listing your eligible mares will come up for you to select your mare. Once you've selected her, click the stud request button. You can also use the request stud link directly on the stallion's page.
There are several reasons for not seeing the "Request stud" link on a stallion.
1. The stallion is not level 10
2. The stallion is not yet 3 years old or is retired.
3. The stallion is starving
4. The stallion does not have the vet checks (breeding check, the regular check and does not have the shots)
5. The stallion is 'currently busy', meaning that it will have to wait for several days. Note: Mules cannot breed.