The balance staff is quite simply the shaft that the balance wheel, the roller table and the hairspring are mounted onto. The two ends of the staff have very small tapered pivots that break easily, particularly if the balance jewels do not have shock springs (like the Incabloc System: see second photo).
If the pivots are only slightly scored, the pivots could be polished using a tool like this Dorrington, without having to replace the balance staff.
The pivot on the left side has obviously broken off and the other pivot appears to be bent:
Since the balance pivot is broken, you must check the balance jewels that these pivots rotate in, to see if a hole jewel or a cap jewel is chipped or cracked. A damaged jewel would need to be replaced. This photo shows a balance hole jewel next to the pallet fork:
After marking the positions of the roller jewel and the hairspring stud on the balance wheel (you could use a felt tip pen that could be cleaned off later), the hairspring is removed by using a pair of miniature tools for this purpose. For clocks and pocket watches, two of the smallest watchmaker's screwdrivers would work well to remove the hairspring, but this must be done very carefully! To remove the roller table, a nice staking set like this one serves the purpose:
Measure the diameter of the pivot. A pivot gauge like this one makes this measurement easy ("12" is 0.12 mm.):
If you have no pivots, you could measure the inside diameter of the balance hole jewels ("0.12" is 0.12 mm.):