"Compensation" is a payment you make to someone for damage you have caused them. ("Compensation" can also mean a payment for work done, like the salary you are paid by your employer is "compensation".)
"Indemnity" has two very distinct meanings. It can mean the same thing as compensation: money paid for damages. But I think the more common meaning is an exemption or protection from having to pay compensation.
So you could say,
When Al damaged Bob's car, Al had to pay Bob compensation of $1000.
When Al damaged Bob's car, Al had to pay Bob an indemnity of $1000.
When Al damaged Bob's car, he did not have to pay any compensation because he had an indemnity clause in his contract.
Oh, just by the way, you "pay compensation" but you "pay AN indemnity". Indemnity takes an article; compensation does not.
"Indemnification" is the process of becoming or the condition of being indemnified. You do NOT "pay an indemnification", you "pay an indemnity". You can say, "Al acquired his indemnification by posting a bond".