Hi Laura, this topic is so neat (I find anyway)!

I know that in many breeds they are now implanting surrogate mares, rather than taking their prized mare out of the show ring. If you use a knowledgeable veterinarian that specializes in equine reproduction, the likely hood of the surrogate mare carrying the foal successfully is very high. This can allow a money earning/sought after mare to have multiple offspring a year, as well as allow mares that aren't healthy enough to carry a foal, but are still highly sought after to still pass on their genetics safely.

A quick run down:

A mare and stallion of choice are mated, or AI is done on the mare of choice.
At this time you want the donor mare to be ovulating as well (this can be induced through drugs).
Roughly 7-8 days after the mare has been bred, they will flush her out to collect the embryos or embryos (7-8 days is the general "good time" to collect the embryo/s from the mare).
You generally want to implant the donor mare within the 1-24hrs, but it is becoming more safe to store the embryo for up to a few months.

That is the quick and simple explanation, embryo transfer in horses is becoming more common, and more safe, but they have been doing this in cattle for longer than horses, so we have some catching up to do in the horse industry!