The tree had to come down, but now what do you do with the stump? You can only saw it so far down and then it's just an ugly block of wood to ruin your lawnmower with and trip over. Burning a stump out takes a little time and a little technique but it is one of the cheapest and easiest ways to go.
Know the condition of the wood you're dealing with. If it's really green, you may have to bite the bullet and wait it out a little bit. Freshly cut wood will make a terrible amount of smoke (if you can even get it to light) and you'll see very little progress with it. Burning is really best suited to a stump that has dried for a while.
Make sure to check with your fire department before you burn to make sure there isn't a burning ban on or a zooming problem that would make burning dangerous, illegal or both. If you go ahead with burning, you will want to take some precautions. Don't burn anything yourself that's close to another building or where there's other flammable materials such as a woodpile, tool shed, or the house itself. Make sure that you have a good ten feet or more between the stump and anything else that could catch fire.
Use a chainsaw to make a really deep "x" in the log, cutting it wide and as deep as you can. Fill the "x" in with kerosene or gasoline and let it sit for a few hours, soaking it in. You can add a little more wood later on if need be. If you don't have a chain saw you can drill very deep holes in the stump here and there from the top of the stump and fill these with kerosene or gasoline.
Remove both ends of a large barrel. A fifty five gallon barrel is perfect. Put the barrel over the stump and toss in some wood and light it. You can use a fireplace log or starter if you have trouble getting it to catch. Let it catch good and then keep throwing wood into it so that the fire stays good and hot. The metal from the barrel will help to keep the fire at its hottest and it will also protect everything around it from catching.
Keep the fire burning for as long as you can. It may burn for two or three days in this fashion. If it stops burning, take the barrel off and see how much you've accomplished. If the stump is still above ground level, repeat the process. When the wood gets below the ground, you can then drill holes into it and pour more kerosene on it to keep it burning deeper. Even if it's burning below the ground, still keep the barrel over it as it keeps it hotter and protects everything around it from flying sparks that could ignite.
Be patient. Stump burning is not the fastest method but it is one of the easiest and definitely the most budget friendly solution.