Jet engines can burn just about any fuel, and most are approved for emergency use of fuels other than kerosene. The maintenance manuals give the approved fuels and maximum number of hours they can be used.
They use kerosene (Jet A, JP-4, JP-5) because it has a high energy content, is easily transportable, remains liquid over a large temperature range, is easily storable, and available world wide. One downside to kerosene is that it is a growth medium for bacteria, and if left sitting for a while, must be treated to kill the bacteria before being used.
Turbine engines use a large amount of fuel (each JT8D on the MD80 uses about 500 gallons an hour in cruise). Jet A is less well-refined than automotive or aviation gasoline, making it cheaper to make (During "normal" times, Jet A is a little more than half the cost of aviation gasoline.)
Because of the large amounts of fuel required for a given flight, a lot of research is being done on alternate fuels, and a Virgin 747 recently made a flight using "biojet" fuel in one of its engines. A company in southern California is working on growing algae that can be converted to jet fuel.