Concrete stains are most often a problem in a garage or basement, but they also can be troublesome on patios and sidewalks. Removing them often requires persistence, but the right materials and methods generally bring success. Try to attack a stain as soon after it happens if possible; the longer a stain sits, the deeper it is absorbed into the concrete or cement. Older or damaged concrete is especially porous and more extreme measures may be required for set-in stains.
Sweep the floor and remove all loose dirt and dust, and then diagnose your problem. Look for oil under the car engines or lawn mowers and snow blowers. Reddish-brown marks are rust, often found where wet snow blowers or garden tools are stored. Plant material (especially maple leaves) leave brown splotches of tannin, a chemical also found in red wine and tea. Mildew looks like what it is---a growing organism---and it may be gray, green or black. Asphalt, tracked by tires or splashed from summer seal-coating projects, is black and shiny (and may be tracked in a tire tread pattern).
Spray and scrub. Most dirt, tannin and other water-based stains can simply be washed away with a bit of scrubbing with detergent using a stiff-bristled natural or nylon (not wire) scrub brush. Use TSP and bleach on mildew. Scrub rust, oil and grease with scouring powder or TSP, then rinse well with water. Use a wallpaper stripper to steam out oil or asphalt stains, then wipe up and scrub with dishwasher detergent or TSP. You also can rent power-washing attachments to clean concrete, but be careful to keep the pressure low. These machines generate up to 6000 pounds per square inch of pressure and can etch, "spall" or take the surface finish off of concrete.
Soak up stains with poultices of TSP and sodium chlorate (available at hardware stores) dishwasher detergent by letting them sit on surfaces to absorb stubborn rust, mildew and oil stains. Scrub the poultice in well and cover with a rag overnight, then wet and remove the next day. Garden lime and cornmeal make good absorption compounds; use them for oil and grease stains. Repeat for stubborn stains. Blot rather than wipe stains when possible to avoid spreading them.
Scrape excess oil and mildew before scrubbing or soaking to make your job a bit easier. Pile ice on fresh asphalt, chewing gum and construction adhesive or caulking; let sit and then scrape as much as possible up with putty knives before reaching for solvents. Latex paint drips will come up well, because the paint forms a skin and dries from the top down rather than soaking in like stains and oil paints.
Save harsh treatments for difficult stains. Use oxalic acid or sodium citrate (citric acid) on oil and grease. Use the type of paint stripper that forms a skin to soften paint and then scrape it off, or try a commercial graffiti cleaner. TSP and many dishwasher detergents contain phosphorous, a strong alkali. It makes them effective, but is very caustic. When using caustic cleaners or toxic solvents, always soak up the stain and dispose of waste according to package directions. Do not rinse these chemicals down the drain or into the storm sewer where they can enter the water supply or waterways.