How to repair grout in a tile floor effectively?

To repair grout in a tile floor effectively, start by removing any damaged grout with a grout saw or knife. Clean the area thoroughly and let it dry. Mix new grout according to the manufacturer's instructions and apply it with a grout float, pressing it into the joints. Wipe off excess grout with a damp sponge and let it cure according to the instructions. Finally, seal the grout to protect it from future damage.