How many foals do horses give birth to at a time?

At best, a mare can have a foal every from age three to age 25, so about 22. Only the very best producers will have more than about ten.
Male horses become sexually mature at around two years of age, and are able to produce a virtually unlimited number of foals. In a wild herd situation, a young stallion usually establishes a harem of mares for the first time when he is 4-5 years old, and is able to maintain a harem until he is about 12-15. Depending on how strong he is, and how well he competes with other stallions, the harem may number between 3 and 30 mares. He will father a foal with most of these mares yearly.

In a domestic situation, where humans control which mares a stallion mates with, the answer varies hugely. He may have no foals at all (if he is not wanted for breeding he will usually be gelded) or he may have hundreds or even thousands - a stallion can mate up to three times a day without losing condition, and modern reproductive technology is able to extend semen so that up to 6 mares can be bred from one ejaculation.

A mare becomes sexually mature at around 3 years of age, but a responsible breeder will not put her in foal until her bones have stopped growing at 5-7 years depending on her breed. Her pregnancies each last a little more than 11 months, and she can be bred again as little as two weeks after foaling. Therefore, she can produce up to one foal a year, and depending on her health, may continue to have foals until she is around 25. Theoretically, therefore, a mare bred every year from the time she is mature until she is too old to breed could have almost 20 foals.
Normally only one a year... though twins do happen occasionally.

A horse can have up to 2 foals, however, twins are very rare-1 in 10,000 and 80% of those die. So live twins are very, very rare.
A healthy, well-managed mare can produce a foal every year into her twenties.