to recompense for something

They gave him ten dollars to compensate him for his trouble.

to counterbalance; offset; be equivalent to

He compensated his homely appearance with great personal charm.

Mechanics . to counterbalance (a force or the like); adjust or construct so as to offset or counterbalance variations or produce equilibrium.

to change the gold content of (a monetary unit) to counterbalance price fluctuations and thereby stabilize its purchasing power.

to provide or be an equivalent; make up; make amends (usually fol. by for )

His occasional courtesies did not compensate for his general rudeness.

Psychology . to develop or employ mechanisms of compensation.