Burrows are user-defined areas where selected dwarves live and work. Dwarves will only perform jobs (use workshops, dig walls, use rooms, etc.) in burrows they are assigned to, though dwarves not assigned to any burrow may still perform jobs located in a burrow assigned to other dwarves.
Military dwarves that are on duty will ignore burrows. Also, burrows do not restrict a dwarf's movements; they only restrict the locations where dwarves may perform jobs (including picking up items, eating, sleeping, etc.). An idle dwarf can still go anywhere, regardless of burrows.
A civilian alert will automatically confine all non-military dwarves to the burrows defined in the alert for as long as the alert is active. When you set the civilian alert level to an alert that includes a defined burrow, your civilian dwarves will ignore their normal burrow-based job restrictions even if their burrow is one of those in the civilian alert.