Burghers were a religious secession group. They broke away from the Church of Scotland (in Ireland that was the Presbyterian church) in 1747. There was a further split in 1799, with some rejoining the main Presbyterian church in 1839, and others (known as the “new lichts”) not rejoining till 1929.
The secession was over the wording of an oath that town officials were required to take which involved swearing loyalty to the true religion. See: