The Quartering Act was a law that Parliament placed over the colonists, requiring them to provide housing and food for British soldiers. The British military commander in America, General Thomas Gage, suggested this act to Parliament. Thus it was passed, much to the dismay of the colonists.
When the provided barracks for the British soldiers were full, then they were able to pick any home or store in the area and just stay there and the colonists weren't allowed to refuse. This, of course, upset many colonists.
Important People involved:
General Thomas Gage