Lots of good answers and definitions here.

Empathy is the ability to understand what it would feel like to feel like another person feels, while still recognizing the difference between understanding what they feel and actually being in their situation yourself. You can have empathy without actually feeling their feelings yourself.

Compassion is the desire to help others.

Empathy is one of the very first and most important things that is taught to students learning to be counselors or therapists. It is essential to be able to understand where someone else is coming from, and still not to lose yourself in their feelings. Compassion motivates therapists to want to help, and is a good thing, but therapists have to be cautious about their compassion and make sure that what they do to try to help is actually in the client's best interest and doesn't encourage too much dependence or other unhealthy behaviors.

Empathy tends to be the basis for good relationships of any kind. Who doesn't need to feel understood? Compassion helps us to look past each other's faults and to forgive and understand that people do the best that they can. Compassion is important in relationships, but again with the caution that we support others and help them in healthy ways that allow them to grow and be strong and become themselves. We need to give what others need to receive, not always what we want to give. And the reverse is true also. We need to receive what is offered, not hold out or reject what is offered because we want something else.