The siege started on the night after the Battle of Lexington and Concord as American forces followed the British back to Boston, and occupied the neck of land extending to the peninsula the city stood on.
At first General Artemas Ward as the head of the Massachusetts militia had the oversight of the siege. He set up his headquarters at Cambridge and positioned his forces at Charlestown Neck, Roxbury, and the Dorchester Heights. At first the 6,000 to 8,000 rebels faced some 4,000 British regulars under General Thomas Gage and had them bottled up in the city.
In traditional terms, the British were not besieged since the Royal Navy controlled the harbor, and supplies did come in by ship. Nevertheless, the town and the army were on short rations. Salt pork was the order of the day, and prices escalated rapidly. Another factor was that the American forces generally had information about what happened in the city, while General Gage had no effective intelligence of rebel activities.
On June 15 the Committee of Safety learned of his plans to attack at Dorchester Heights and the Base of the Charlestown Penninsula. They sent word to General Ward to fortify Bunker Hill and the heights. He assigned Colonel William Prescott the Bunker Hill task. On June 17 British forces, under General Howe seized the Charlestown peninsula at the Battle of Bunker Hill. They did take their objective, but didn't beak out since the Americans held the ground at the base of the peninsula. British losses were so heavy that there were no more direct attacks on American Forces. From this point, the siege became essentially as stalemate.
On July 3 George Washington arrived to take charge of the new Continental Army. Forces and supplies came in from as far away as Maryland. Trenches were built at the Dorchester Neck, and extended toward Boston. Washington reoccupied Bunker Hill and Breeds Hill without opposition. These activities had little effect on the British occupation.
Then, in the winter of 1775-1776 Henry Knox and his engineers used sledges to retrieve the heavy cannons that had been captured at Fort Ticonderoga. Bringing them across the frozen Connecticut River, they arrived back at Boston in March. Now the British fleet ceased to be an asset. It was anchored in a shallow harbor with limited maneuverability and under the American guns on Dorchester Heights, which General Thomas had fortified. The siege was over when the British set sail for Halifax on March 17, 1776. The militia went home and in April Washington took most of the Continental Army forces to fortify New York City.