The Quartering Acts refer to two separate acts passed by the British Parliament to ensure that British troops would be properly billeted and fed during their times of service in the North American Colonies. The first act was passed in 1765, the second in 1774.
The first act, passed on March 24, 1765, simply instructed the colonial governors to ensure that soldiers were provided with adequate food and housing. If sufficient barracks and public houses were not available, soldiers could be billeted in commercial properties such as warehouses, in empty homes, and in barns. Much as today, the populace was mis-informed and perfectly willing to be so, and it was commonly believed that this act allowed soldiers to be quartered in private homes.
However, this was an obnoxious act in other ways; it required that the Colonial authorities pay for the housing, food, and drink for troops without compensation, and did not make allowances for the local circumstances. This was also accompanied by an unexplained increase in military presence. No standing army had been maintained in the colonies before the French and Indian War, and the colonists were not happy to find that they were still to be 'occupied' once the French were defeated. Some leaders, most notably the New York Provincial Assembly, refused to abide by the terms of the Quartering Act. This left 1,500 British troops stuck on their ships in New York harbor with nowhere to go, which caused some upset in England. The first Quartering Act expired on March 24, 1767, which eventually led to the 1774 Quartering Act.
This second act only addressed housing, not food and drink. As with the previous act, it did not provide the governor the right to billet troops in private homes. The act also specified that the housing be arranged through mutual agreement between the parties involved, and provided punishments for cases in which troops were housed inappropriately in private homes. However, this act was only one of a flurry of acts passed after the Boston Tea Party, which were referred to as a group as the Intolerable Acts. The Quartering Act was seen as an indirect tax on the Colonies, requiring the colonists to help support a standing military that they did not want.
The Quartering Acts were mentioned in the Declaration of Independence and are part of the reason that the 3rd Amendment was included in the American Constitution. However, the actual housing of British troops in American homes took place during the French and Indian War, before any act was in place. Ironically, the 3rd amendment still allows for the possibility of housing soldiers in private homes in times of war -- as long as it is in accordance with the law.