"Empathy" is the state of seeing yourself in somebody else's place, and feeling their emotions as if they were your own. The emotions can be either happy or sad.
Example of use:
"I can emphasize with my granddaughter's anxiety about entering college, because I know that it was like when I was entering college."
"Sympathy" is the state of being aware of someone else's emotions and thinking they are important, but not experiencing them yourself. Usually it is used in the case of sad emotions.
Example of use--a condolence note for someone whose child has died:
"I feel sympathy for your loss. I've never lost a child myself and I can't even imagine what you must be going through."