Let's see...you were changing the oil on your lawnmower (or worse, your car) and the old oil somehow missed the catch pan without you noticing. Isn't it amazing how oil spreads? The best thing to do is not to panic, but to act fast. Contain and soak up the oil first, then hit the stain with a strong cleaner.
Try any or all the following, depending on what you have on hand. And for next time, it's a good idea to buy some concentrated natural soap (the first solution on the list). Also, don't be so lazy next time—lay down some cardboard before changing your oil.
Soak It Up ASAP!
Start tackling fresh oil stains by blotting up as much oil as possible with paper towels or rags. Just lay the towels over the spill, let the oil soak in for a few seconds, and pick up the towels. Don't try to rub the oil "out" because you'll only rub it in, or spread it.
The right soap in the right concentration can be a miracle oil-stain remover—along with some serious scrubbing, that is. This is not a product endorsement, but one soap that will truly remove oil from concrete is Mrs. Meyer's Clean Day Multi-Surface Concentrate. It's a natural-ingredient liquid soap sold in big 32-ounce bottles and is usually diluted for general household cleaning. But in this case, apply it full-strength directly to the stain, then scrub vigorously with a stiff nylon brush, like one of those big iron-style tub brushes with a handle.
Rinse the area with a garden hose and nozzle, using a strong stream, if possible. Repeat one or two times, and chances are the stain will be gone. You can tell the oil is gone when the rinse water no longer beads or "rainbows" on the concrete. Other natural soaps may work just as well, but they probably have to be concentrated, as in this case. At full strength, Mrs. Meyer's concentrate is strong stuff, and the vapors will sting your eyes a bit, but it works.
Sprinkle inexpensive absorptive cat litter on the stain, making sure to cover the oil stain thoroughly. Crush and grind the litter into the oil stain with your shoes. Let it sit for an hour, then sweep it up and discard it. For very heavy stains, let the kitty litter set overnight after crushing, then sweep and dispose. Wash the area with a heavy-duty detergent soap (or something like Mrs. Meyer's) and a stiff brush, then rinse. Repeat as necessary.
Add 1 cup of phosphate-free TSP (trisodium phosphate), also called "TSP-PF," to a gallon of hot water, and mix. Wearing eye protection and rubber gloves, pour the TSP onto the oil stain and let it soak in for 20 to 30 minutes. Then, scrub with a stiff nylon brush, and rinse with a hose. Repeat as needed.
Paint Thinner and Sawdust
Mix sawdust with paint thinner until it is damp, then spread the soaked sawdust over the stain. Let it soak into the oil stain for 20 minutes. Sweep up the sawdust and repeat, if necessary.
Microbial Oil Stain Removal
Use a biodegradable microbial oil stain remover, such as EATOILS BT200™, which uses micro-organisms to actually eat the oil and grease away.