Actual physical damage to stair treads is rare and will probably be limited purely to split or broken nosings. These can be repaired by cutting them off flush with the riser below, using a chisel, and pinning on a new molding.
A much more common problem, particularly in older houses, is creaking as a result of the treads becoming loose. The ease with which this can be fixed depends very much on whether you can get to the underside of the stairs or not. If you can, simply pin and glue 2 x 2in triangular blocks of wood between the treads and risers below, and drive screws up through the tread into the riser above. This is the only way you can fix a staircase with closed strings.
If the staircase has one or two cut strings, you can make the repair from above. First prise off the molding from below the tread nose and the molding holding the foot of the baluster in place, using an old chisel. Run a hacksaw blade along the gap between the back of the tread and upper riser. cutting through any fixings. Alternatively, cut through the riser itself with a backsaw. Drive a chisel blade between the tread nose and riser and lever it free. Then you can remove the risers if damaged.
If necessary. cut a new tread and riser from wood of the same size as the originals.
If one of the strings is closed, glue and pin supporting blocks to it for the ends of the riser and tread. Use offcuts of the tread and riser wood as positioning guides to ensure a tight fit. Then glue and pin the riser in place.
Pin and glue more blocks to the top of the lower riser and then glue the tread on top, strengthening the bond by driving screws or nails down through the ends into the cut string or strings. Do not drive any screws or nails through the leading edge of the tread as they may become exposed as the tread wears.
Refit the baluster, pinning it to the handrail and then pin the retaining molding to the end of the tread. Finally, refit the molding beneath the tread nose.
If you can reach the underside of a closed string staircase, you can replace treads or risers by removing their retaining wedges with a chisel and sliding the damaged parts out. Slot the new pieces in and fit new wedges. If a carriage runs down the centre of the stairs, however, the work is best left to a joiner or builder.