A fob is a chain that holds an old-fashioned pocket watch. If your grandfather carries his watch around in a little pocket in his vest, it's probably attached with a fob.
It's uncommon to see a watch fob these days, but they were once as ordinary as the watches they held. The little pockets sewn in men's clothing were also fobs, and today, a key chain is sometimes called a fob. When fob is a verb, it means to fool people or play a trick on them, especially by giving them something less valuable than they expect. This meaning is from the obsolete fobbe, "cheater."