Baptists are named for a rite, and the rite is a proxy for a bigger claim about what a church is.
The tradition begins with English Separatists, congregations that had given up on reforming the Church of England from within, among them a group in exile in Amsterdam under John Smyth, who around 1609 concluded that infant baptism was no baptism at all and baptised himself and then his congregation. Thomas Helwys brought part of that congregation back to England around 1612 and wrote one of the earliest English arguments for complete liberty of conscience, including for Catholics, Jews and Muslims, which was an extraordinary position at the time and got him imprisoned.
Two Baptist streams formed. The General Baptists were Arminian, holding that Christ died for all; the Particular Baptists, from the 1630s, were Calvinist, holding a definite atonement, and produced the London Confession of 1644 and its influential 1689 successor. The family has contained both ever since, which is one reason Baptists are so hard to summarise doctrinally.
The theological point beneath the water is this. If baptism is administered to infants, the church is coextensive with the community, and everyone born into the parish is in it. If baptism follows a personal profession, the church is a gathered body of the convinced, and membership means something a person has done. Everything else Baptists are known for follows: the insistence that the local congregation is the church; the refusal of any authority above it; and the demand for separation of church and state, which Baptists held long before it was respectable and which they argued for as a religious principle rather than a secular one.
Baptism by full immersion came somewhat later than the principle itself, from the 1640s.