You will deal with breeding requests as both a mare owner and a stallion owner. If you have a mare, you need to go to the Stud Book or Genotype Search (in the Breeding tab) to look for available stallions. From there, you can place a request for any stallion who is not marked as full. Requests may be placed as early as July of each year in preparation for the following year's breeding season. Placing a request does NOT guarantee that the mare will be able to breed to that stallion. You'll have to wait and see if the stallion owner accepts the request, which they cannot do before December.
You can visit your Mare Services page to view the status of requests you have made for your mares at any time. This will tell you if you have a request made for a mare, who it is to, and if it has been accepted yet. You can also delete requests from this page if you've decided you no longer want to go through with it. However, it is considered rude to delete a request once it has been accepted without at least notifying the stud owner. When they accept a mare, they are marking off a spot in the stallion's book for her. If you don't use that spot, it results in lost income for them and could potentially keep someone else from getting to take over that slot. So if you do decide you no longer like the match, at least send the stud owner a note to let them know they can offer the slot to someone else.
You can list stallions for stud (or edit their fees) beginning in July of each year from either their Manage page or your Stud Services. Requests will begin coming in soon after, and while you can deny them at any time, you cannot officially accept any until December. This is to prevent breeding from simply becoming a race to request to the best studs - do yourself and everyone else a favor and give them time to get all of their mares matched. Once December rolls around, you have the option to accept or deny any request you want, for whatever reason you want. You should always be thorough and check to make sure the breeding looks like a good match. At the very least, take a glance at the genetics and make sure there aren't any glaring problems (like crossing a fair x good Intelligence or a bloodline I to V).
If a stallion owner chooses to accept a request, the mare will be able to breed once breeding season begins in February. If a stallion owner denies the request, you will be able to make another request to a different stallion. Please don't harass them if they don't accept. Their stallion may be swamped with requests, they may have had better mares request, or they may just not like the genetic match. The bottom line is that it is their choice, and you should be polite about it. You can always ask for a clarification if you want to find out how you can improve matches in the future, but please accept their decision as final.