From Wikipedia entry on Leicester Square:

The Square is named after Robert Sidney, 2nd Earl of Leicester, who purchased four acres (1.6 hectares) in St. Martin’s Field in 1630; by 1635, he had built himself a large house, Leicester House, at the northern end. The area in front of the house was then enclosed, depriving inhabitants of St. Martin’s Parish of their right to use the previously common land. The parishioners appealed to King Charles I, and he appointed three members of the Privy Council to arbitrate. Lord Leicester was ordered to keep part of his land (thereafter known as Leicester Field and later as Leicester Square) open for the parishioners.

The area was developed in the 1670s. It was initially fashionable and Leicester House was once residence of Frederick, Prince of Wales but by the late 18th century, the Square was no longer a smart address and began to serve as a venue for popular entertainments. Leicester House became home of a museum of natural curiosities called the Holophusikon in the 1780s and was demolished about 1791-1792.