1. The Hammer
This method can work of the starter solenoid is sticking. You can tell this if when you turn the key you hear a click but nothing else (no normal starting noise) , but have a good battery. Go get your self a stick, hammer, tire iron, large screwdriver, or any other long object that you can whack the starter with. Don't hit it as hard as you can as this my break your starter, but start off fairly light, it shouldn't leave a mark on the old starter, then try to start it. If it doesn't work give it another slightly harder whack and repeat. If after a few tries it still doesn't start you will need to replace the starter.
2. Jiggle the car
This method can work if you have a worn gear on the starter. You can tell this if when you try to start the car you hear a high speed motor spinning. It sounds similar, but quieter, to a vacuum cleaner. This noise is the starter motor not catching the gear on the outer edge of the flywheel on the engine. Put the car in gear and try to move it back and forth a couple of times. This may move the gear on the flywheel enough so that the starter can grab it now and start the car.
3. Hot wire it
This method can work if your solenoid has completely failed, you don't even hear the click mentioned in number 1, or you have bad wiring going to the solenoid or starter. It is easiest on vehicles that have the solenoid separate from the starter but might be possible on vehicles with the starter and solenoid all in one. Here you will need to turn the key to the run position and then create a short across the terminals of the solenoid. If you have an external solenoid this is easier as you can see what needs to be shorted, it will be the terminals with the really big wires connected. I suggest using a large plastic handled screwdriver for this as they can carry the very large current and you won't electrocute yourself.
4. Pop start it
If none of the above methods work and you have a vehicle with a manual transmission this will work, provided you can get the car rolling fast enough. Again first you will turn the key to the run position. Then put the vehicle in neutral. Next you will either need to get out and push the vehicle, or let it roll down a hill. When the vehicle is going near 5 mph get back in, close the door, put it in gear and let the clutch out and be ready to give it some gas. If it doesn't start you will have to try again, I suggest getting it going faster. The only time this hasn't worked for me on a manual transmission vehicle when when I had to push it up a hill.
These tricks may not work every time but I have used all of them in the past with success. When you do finally get your vehicle running remember you are on borrowed time and need to replace your starter, so I suggest driving to the auto parts store leaving your vehicle running and getting a new starter as you may not be able to start it once turned off. If none of these are an option then you will be stuck either paying for a tow or borrowing a vehicle to get a new starter.