The horse, unlike the sheep or goat, is designed to produce one offspring at a time. Twinning does occur, but it's generally considered a very bad thing. The most common outcome is for only one of the foals to survive until birth, but it's not at all uncommon to lose both foals - and sometimes the mare as well. The chances of both surviving and being healthy? About 1 in 10,000 - and even then, they tend to be smaller and weaker than singleton foals.
Because of this, with modern technology, "reduction" is normally performed - this first happened in the 1980s but only became common in 2006. An ultrasound can detect the twin pregnancy, and then reduction is performed at the same time - it's also called "pinching" the extra foal. They always try to reduce the smaller of the fetuses. If a reduction isn't possible it's not uncommon to just use an abortion drug and try again. (And if a reduction is done incorrectly, you can end up losing both foals. Drugs are given to the mare afterwards to help her hold on to the remaining fetus).
Before these techniques were available, though, a twin pregnancy was a horse breeder's dread because of the major complications that ensued. Twinning is more common in mares that have their first foal when older, and some mares are prone to it.
There are, incidentally, numerous reported incidents of mares raising and nursing multiple foals, including spontaneous adoption of orphans within the same herd. So if you see a horse with "twins" it's likely one of them is not biologically hers.