The Act that is being described is the Quartering Act, which was passed by the Parliament of Great Britain on May 15, 1765, at the request of Lieutenant-General Thomas Gage, commander-in-chief of forces in British North America. By this act of the Parliament, the local governments of the American colonies were forced to offer housing and provisions to British soldiers that had fought in the French and Indian War (1754-1763). If barracks and public houses were not enough to house the soldiers, they would be quartered in other places, such as inns, as well as in the houses of people that sold alcoholic drinks, and colonists must pay for their expenses.
A second Quartering Act was passed a few years later, in June of 1774, but it did not require that troops be supplied with provisions, and a mutual agreement between the parties involved was necessary before proceeding to house the soldiers.