The fuel used by the main engines on a Falcon 9 is liquid oxygen (LOX) and Rocket Propellant 1 (RP-1), which is a highly refined kerosene, similar to JP-5 and JP-8 jet fuel. The attitude thrusters, used to turn the stage around for the boost-back burn, are described in SpaceX literature as "cold gas thrusters", and are much like a CO2 fire extinguisher.
The Dragon capsule, which is sometimes the cargo on a Falcon 9, does use hydrazine.
Elon Musk is somewhat of an environmental nut (hence the electric car company), and the next rocket they are working on will use liquid natural gas instead of kerosene as the fuel. It's not only cleaner burning, but is easier on the engines, which are intended to be re-usable. The kerosene leaves some soot on the engines, which must be cleaned off before re-use.
The Russian Proton rocket and the Chinese Long March 2F are the only major launch vehicles that use hjydrazine (unsymmetrical dimethylhydrazine) as the main fuel, and nitrogen tetroxide as the oxydizer. They make a real mess if they crash. The Russians are phasing out the Proton, replacing it with the Angara rocket, which will burn kerosene. Of course, nobody knows what the Chinese are doing.
The two large Russian modules of the International Space Station were launched on Proton rockets.