Branding can serve a few different purposes. A free man or woman might accept a brand as part of swearing an oath, of allegiance or otherwise.
(That includes marriage vows.)
For a criminal, a brand is a mark of ones sins - others who see it may well think poorly of you and shun you. It's basically a primitive form of criminal record, one which can follow an offender wherever they go in the day's before coordinated police forces. Which means trouble if a branded criminal is caught committing another crime, or even just in the area where a crime of that sort was committed lately - a second offence may carry a far more serious punishment.
Slaves can also carry punishment brands of course - like the mark of a runaway (generally on the face or somewhere else obvious) or a thief. Although some masters prefer to avoid such things - since they lower the sale value. But also common among slaves is the ownership brand - on some slaves, several, as like with marriage, some masters commit a bit soon.