A beau is an old-fashioned term for "boyfriend." When your great-grandmother was young, she probably had a beau.
Beau means "handsome" in French. Pronounce it the same way the French do, with a long o: "boh." The word recalls more innocent times, when gentleman courted their ladies and skirts barely rose above ankle level. You'll hardly ever hear beau used today, unless you're watching a movie, play or television drama set in the past.