If you're imagining the kind of pilgrim who landed at Plymouth Rock, you're… well, close. Bunyan was an important figure in the 17th-century Puritan community, which, due to persecution in England, moved in large numbers to America. Some of 'em came over to North America on the Mayflower. The term pilgrim, however, goes back much farther.
Since the 11th century, travelers, wanderers, and particularly those on spiritual journeys have been known as pilgrims. Just think of the travelers in The Canterbury Tales or even people who go on "pilgrimages" today to Jerusalem, Mecca, or even Graceland.
Calling his story "The Pilgrim's Progress," Bunyan is generalizing in an important way. The "progress," or journey, that he's going to narrate is every man's journey who sets out on a pilgrimage. But he's especially shouting out to those who are searching for Christ and trying to act as good Christians.