This building housed the Massachusetts government from 1713 to 1798. Before the revolution, that would have meant the governor, his advisors, and the elected assembly. Of course, by 'elected', I mean 'selected by white male property-owners' since they were the only people that had the right to vote.
The assembly also had rather limited power - the governor had a veto over anything they decided, and on occasion dissolved the assembly when they didn't vote his way. Less a governor, more a dictator.
In 1767 a gallery was installed above the meeting floor here so the general public could watch their legislature at work - this was a very novel idea at the time. Of course, some of the representatives probably regretted the decision later, when crowds used the area to heckle those who voted pro-British. Then again, perhaps that was the point all along.
The government moved out of the building to a bigger space in 1798 and the old building was turned over to merchants, including a wine seller and a wig-maker, making it the colonial equivalent of a shopping mall, though one that mainly dealt in wine and wigs.