There is no easy way to remove concrete paint from floors, but with some paint stripper and some hard work, you can bring even a floor covered with many layers of paint back to its original glory.
Clean the floor with trisodium phosphate cleaner (TSP) before starting the stripping process if there is grease or a lot of dirt on the floor. A very dirty or greasy floor can hamper the efficiency of the chemical paint stripper.
Read and observe all warnings and directions on the paint stripper label. This is a toxic and often flammable substance. Some brands are nonflammable, although they may not be quite as effective. In some cases it may be illegal to dispose of the old paint and residue from the stripping process with the regular trash. Check with your waste removal company and have some heavy duty contractor trash bags on hand for the waste product.
Make sure the area you are working in is well ventilated. If working in an enclosed space like a basement, open windows and use fans. Turn off pilot lights and anything else that has a flame or can cause a spark, and do not smoke (unless you are using a nonflammable paint stripper).
Use a large, stiff-bladed putty knife to remove as much of the dissolved paint as possible. Check with your local hardware store for scrapers with long handles as this will make the job easier on you. Clean up by dumping the scraped up paint in a trash bag as you work.
Use the wire brush on the end of an extension pole or broom handle to get the final residue off the floor. You will probably have to go over at least some areas more than once, depending how many coats of paint you have to remove.
Powerwash, if possible. While this won't be an option when working inside a house, powerwashing a garage or patio floor will speed the process. Be careful about soaking sheetrocked walls but if you direct the water away from the walls and use the water pressure to lift away much of the old paint you can save a lot of time.
Wash the floor with a strong solution of TSP after all the paint is removed, and rinse well. This will get up the chemical stripper residue and any other grease or dirt that has been long hidden by the paint.
For extremely stubborn paint film or if it is simply not possible to use chemical strippers on the floor, check rental yards for mechanical concrete scrubbers or floor grinders. Be aware that these machines can also remove some of the concrete and are big and noisy to use.