When someone reimburses you they pay you for the cost of something. Essentially they give you money that they owe you.
It is a form of compensation. For example:
- Sorry I broke the lamp. I promise to reimburse you (pay for it / get another one)
- The investors should be reimbursed for their losses after his incompetence.
It could be an expected compensation though:
- Your expenses will be reimbursed (if you have to pay for something to do the job you will be paid back)
Compensation itself is more general. It could be a way to pay someone back - like the examples above - or it could be additional payments for perceived losses as well. These might include the stress you suffered, hassle you went through, etc.
So, when someone deserves compensation for something - when they are compensated - it is usually a mix of being reimbursed for the expense they suffered and sometimes some additional money for the hassle/hardship caused.