Mix a solution of oxygen bleach and water in a bucket. Exact proportions vary from brand to brand, but 1 cup of oxygen bleach powder to 1 gallon of water is usually a good starting point.
Pour a small amount of the solution onto an area of the tile floor. Spread it evenly with the mop, but don't attempt to scrub yet. Continue pouring and spreading until the entire floor is covered with the solution.
Wait 30 minutes, then return with the mop. Dip the mop into the bucket, wring it out and scrub the floor, section by section, until the entire floor has been mopped. Use a scrub brush on stubborn stains.
Rinse the floor with clean water, applied with a mop, and dry up excess water with clean rags or shop towels.