Energy is measured in Joules, J. It does not matter what type of energy it is, it always has this unit.
As a general trivia, Joules is not in fact the fundamental SI unit for it, that's [tex] \frac{kgm^2}{s^2} [/tex]
This is because w=fd where f is the force in newtons and d is distance
f also = ma where m is the mass and a is the acceleration
Putting these together, w=mad (it's mad right...), and so replacing the values with the units we can say
[tex]J=kg* \frac{m}{s^2}*m= \frac{kgm^2}{s^2} [/tex]
