An empathetic person is someone who can share another person's feelings. If you tell an empathetic person that your heart is broken, she might touch her own heart and gaze at you sadly through moist eyes.
Empathetic is a recent term; it comes from empathy, which was coined by the German philosopher Rudolf Lotze in 1858. Lotze believed that when you look at a work of art, you project your own sensibilities onto it. So if you feel sad when you see a painting of a woman weeping over a dying lover, that's because you can imagine what it's like to lose someone you love. To make the word, Lotze turned to the ancient Greek empatheia, which means "passion."