The Bounders were responsible for policing the borders of the Shire and ensuring that outsiders behaved themselves. This was known as 'beating the bounds'. The Bounders was the larger of the two branches of the Watch (the other being the twelve Shirriffs).
The number of Bounders at any given time was not fixed but varied according to need. In the run up to the War of the Ring they were greatly increased as ruffians and spies of Isengard and Mordor began to enter Eriador. Unbeknownst to the Hobbits of the Shire this was the first sign that the troubles of the south would soon reach even their borders.
The word means "persons watching the boundaries".