You will have access to a Genotype Search on the Breeding tab. This is fairly self-explanatory once you begin using it. The search includes quite a few parameters you can select: surface, distance, and all of the genes. You can choose whether you want to include a search for specific hidden genes, or if you just want to look for the normal genes
This is probably one of the best features to have if you are a large stable and want to get involved in breeding a lot of mares each year. While it can provide you with a long (or short) list of stallions that meet your criteria, don't forget to hypomate these stallions still just to get a visual of how they match your mare. Hypomating may also help you narrow down the options if a lot of studs meet your requirements.
You can also use the genotype search to find mares rather than stallions. This will only search within your own mares. You can use it to do some reverse matching, finding mares to match stallions you need to fill.
Did you not get any results, or not any you like? Try broadening your search parameters. For example, it's usually best to start out by trying to fix any red genes your mare has, and any poor/fair stat genes. If that didn't turn up many options, you may want to change the red genes to just be an Xx Xx pair over all. For example if you first searched for __ DD for Development, you could change it to Dd Dd. Sure, if you cross Dd Dd by Dd dd, you might end up with a low development foal - but ultimately that risk could be better than giving up on other things (like fixing low stat genes).
If you're unsure of exactly what you should look for to improve your mare, swing by the Live Chat and ask some veterans for help! They are very friendly and more than willing to help you get started in the breeding side of the game. You'll be much happier if you ask for help and end up finding a great match than if you just choose a random stallion and end up with a poor foal as a result.