The New England Patriots were founded in 1959 as the Boston Patriots, joining the then-new American Football League.

Quick Scoop

Exact founding detail

Most official histories place the team’s founding in 1959, when Boston businessman Billy Sullivan was awarded an AFL franchise and established the Boston Patriots, who later became the New England Patriots.

In simple terms:
The New England Patriots trace their origin back to 1959, when the Boston Patriots franchise was created in the AFL.

TL;DR: The New England Patriots were founded in 1959 as the Boston Patriots.