Baptists are essentially an offshoot of the English Separatist movement, which was itself the radical wing of Puritanism. The Puritans were English Protestants who thought the Church of England needed to be reformed in a more radically Protestant direction. The Separatists decided that this was impossible because Anglicanism was a “daughter of the whore” (the whore being Catholicism!) and thus irreformable. They began establishing their own independent congregations at the end of the 16th century, and some of them (including some early Baptists) were burned at the stake as heretics by the Anglican government. Because of this vicious persecution, many Separatists fled to the Netherlands (you probably know this story if you are an American, because some of them then migrated to North America and landed on Plymouth Rock–and now you know the rest of the story!). Some of these emigres decided that infant baptism was un-biblical. Wardrandolph is right that one of the first of these was John Smith. In the Netherlands there were quite a few Mennonites–adherents of the 16th-century Anabaptist movement, which also rejected infant baptism. So many Baptists claim to be descended from the Anabaptists (and some fundamentalist Baptists go further and make the absurd claim that there is a “trail of blood” of Baptists throughout Christian history). There may well have been some influence, but from what I’ve read the evidence seems to point toward the Puritan/Separatist heritage being more important. For instance, John Smith roomed with a Mennonite in the Netherlands. Maybe the Mennonite helped convince him that he should be rebaptized, but the fact is that Smith did not ask the Mennonite to baptize him–he baptized himself, because he didn’t think he knew any validly baptized people whom he could ask to perform the ceremony!
Therefore, Baptists are best seen as radical Puritans/Separatists with perhaps some Anabaptist influence. The Baptists were divided into two groups–Particular Baptists, who were Calvinists with respect to predestination, and General Baptists, who were not (and for whom the Anabaptist influence is more probable than for the Particulars). Most Baptists in the U.S. are derived from the Particulars, but the majority of them have watered down their Calvinism to a belief in “eternal security.” However, Calvinism seems to be on the rise in some Baptist circles.
The best source for Baptist history I know of is Leon McBeth, Baptist Heritage: Four Centuries of Baptist Witness.
Edwin