When you sympathize, you care about how someone else feels during a time of trouble. If you sympathize with your brother, whose best friend is moving away, you understand why he's feeling sad.
When you sympathize, you care, so you might sympathize with the victims of an earthquake in a foreign country or a child who's waiting for her turn on a merry-go-round. Sympathize comes from the Greek sympathes, have a fellow feeling for. It's sometimes confused with empathize, which means putting yourself in someone else's shoes, rather than simply caring about them.