Clean the floor with a mild cleaning agent such as TSP (tri-sodium phosphate). Open windows and set up fans to ventilate the area and wear rubber gloves, a surgical mask and goggles as necessary.
Use a putty knife to remove drops of paint or an older coat of paint. If this proves difficult, try the wire brush. In the case of wood floors, take care not to damage the floor itself.
Apply paint thinner, following the manufacturer's instructions. This may require the use of brushes or rags to rub it out of the floor. If this fails, try acetone.
Buy or rent a floor sander. This powerful tool can can easily damage the underlying floor, so be conservative with its use. In the case of concrete floors, consider using the old paint as a base coat if it's in decent condition, as sanding can greatly damage the texture and consistency of the floor.
Repeat each step as necessary and try material-specific cleaning products as well. Consider hiring a professional if problems arise or if the old paint cannot be easily removed.