Fixing a broken staircase step can be a simple task. Sometimes is requires the help of a professional. The process of fixing a broken steps and whether or not you will need professional help depends on how your staircase was originally built.
If the tread is secured to the top of the stringers and there are no handrail balusters attached to the tread, you should be able to easily remove the tread. Either pry it up, or unscrew it. Lift the broken board away. Replace it with a new wooden tread.
If your staircase was constructed with mortises on the stringers with the treads glued in place, removing the tread can do considerable damage. You may need to cut away to the top board, leaving the mortise joint in place. Screw down a flat tread board in its place.
If the broken staircase tread has a baluster attached, it is probably mortised into place. It can make removing the tread a complicated operation, which best left to a professional. You'll need to remove the baluster before you can remove the tread, which could cause more damage to the staircase. If you remove the baluster and the tread, you'll have to cut the proper mortise joint into the replacement tread for your staircase.