My understanding of EMPATHY and COMPASSION.
(As already mentioned by vajrastorm and biggyboy)
For a person to have compassion toward others, one first must have EMPATHY.
The ability to recognize the feelings of another being, given a certain condition or situation.
To put oneself in another's shoes and feel how that person/being is feeling.
EMPATHY is a kind of understanding of feelings of another being.
By understanding another being's pain and suffering and a DESIRE to help or ease it, is EMPATHY.
COMPASSION, is very similar to EMPATHY I think, but it goes a little deeper than that.
To have COMPASSION, one FEELS the pain and suffering of another being (EMPATHY), and take the extra step forward regardless if it's convenient or not without any personal agenda, to do something to stop, if not lessen the other being's problem.
An example:-
We see a handicapped beggar going around begging for money to survive. We immediately feel very sad and painful for that person. Probably out of kindness and the EMPATHY we give that person a dollar or two, in hope that it will "temporarily" ease his/her problem, and pray for a better tomorrow for him/her after he/she leave us and continue to beg from another person. And then we continue on with our own life soon forgetting this poor beggar.
A person out of COMPASSION and genuine care (not saying EMPATHY is bad or fake but just to show COMPASSION goes a little further), when given the same situation will do a little more. Probably take the extra step to understand how he/she got into such plight, check with the beggar where he/she is living, regularly visit and help that person, try to improve the living condition of that person by getting help from welfare departments or public donations, make sure he/she is provided for and assist him/her to find a job so that he/she can be independent and need not beg anymore in the future. Really going all the way to help another.
I think that is the difference between EMPATHY and COMPASSION.