The trick is called the "burning" note trick.
This is a good 'magic trick' that illustrates the process of combustion, the flammability of alcohol, and the special qualities of the material used to make currency.
A combustion reaction occurs between alcohol and oxygen, producing heat and light (energy) and carbon dioxide and water.
When the bill is soaked an alcohol-water solution, the alcohol has a high vapor pressure and is mainly on the outside of the material (a bill is more like fabric than paper, which is nice, if you've ever accidentally washed one). When the bill is lit, the alcohol is what actually burns.
The temperature at which the alcohol burns is not high enough to evaporate the water, which has a high specific heat, so the bill remains wet and isn't able to catch fire on its own. After the alcohol has burned, the flame goes out, leaving a slightly damp dollar bill.
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