A sympathetic person is one who's motivated by compassion. You can imagine that most of the people who work for the Red Cross are sympathetic types.
In literature and film, the sympathetic character is the one who is likable or who evokes feelings of sympathy from the audience. The sympathetic character in a romantic comedy is probably the romantic male lead who keeps getting ignored by the beautiful women. Eventually the sympathetic character will get noticed and the leading lady will fall in love with him, then they will live happily ever after.