I can give you some information on what a burgess is, which is essentially the same thing as a burgher, a citizen of a town or borough, but the English version, and I only really know monastic towns well enough so I will stick to English monastic towns to try and give you an answer.

Life as a burgess in a monastic town would have varied to that of a town under the government of a secular lord. Throughout the middle ages, burgesses wanted more and more liberties and freedoms from their lords, monastic or secular. The monastic lords were more set in their ways, and did not want to give up their rights to their towns. Leading up to and including the Peasants' Revolt in 1381 in England, the burgesses were constantly fighting for more liberties, which could include being able to use their own hand mill to grind their grain, being able to elect their own MPs and being able to elect their own mayor. The Peasants' Revolt involved the townsmen (burgesses) rising up against their lords, forcing them to write charters to give them their wanted liberties, and in some cases murdering them. Generally, the Peasants' Revolt was unsuccessful for the burgesses.

In monastic towns, the monastery tended to hold manors, which they would have their peasants (serfs) worked on. The peasants would work the land, and depending on the system of farming used, they could keep some of the produce to sell or eat, and some would go to the lord of the manor to feed the lord. The serfs often owed their lord services as part of their tenancy, which could include attending a court. There were differences between free and unfree tenants as well. Unfree tenants wouldn't have even been allowed to marry their daughter without the permission of their landlord, and certainly wouldn't have been able to leave the land without permission.

It is a very confusing topic when getting into the details, and of course varies drastically between the earlier Middle Ages and later on, my answer is leaving towards later!! I hope I have helped a bit and haven't confused you. If you want more details and have more questions please ask!!

Sources -

Susan Reynolds, Introduction to the History of English Medieval Towns (1977) and Kingdoms and Communities (1997)

Rodney Hilton, The decline of Serfdom (1969) and English Peasantry in the Later Middle Ages (1975)