What is the name for a mother horse?

A female horse is called a 'Mare' but you can also use the term 'Dam' to refer to a mother horse because 'dam' is a female parent (used especially of four-footed domestic animals).

A male horse is called a 'Stallion' but you can also use the term 'Sire' to refer to a father horse because a 'sire' is a male parent (used especially of four-footed domestic animals).