Rumor is unconfirmed or unverified information. For example, "I heard that the factory is going to close down at the end of the year!" That is a rumor.
Gossip is similar, but it is usually "secret" stuff about somebody else. For instance, a co-worker whispering, "Did you know Jane is pregnant?" That's gossip.
Sometimes the two words can be used interchangeably; there is some overlap in their meanings. However, generally speaking, a "rumor" is nothing to be ashamed of; it's just unconfirmed information. "Gossip" is usually about a person(s) in particular, and causes embarrassment if that person(s) finds out that somebody was spreading the gossip.