A carbon arc lamp is a specific kind of arc lamp where the electrodes are carbon rods and the gas used between the rods is air. Follow the link below if you don't know what a general arc lamp is.
Carbon arc is an old welding method which uses relatively low voltage, high amperage electricity to heat metal when an arc is established between a carbon electrode and the work (single-carbon arc)...
You have two electrodes mad out of carbon, and then you cause a sparc to arc over from one electrode to the other. By maintaining this arcing you get light.