The answer immediately below omits (among possibly others) the 1920 Akron Pros, who went 8-0-3 and were crowned the NFL's first champions.
The answer below omits the 1929 Green Bay Packers who were the league's first undefeated champions (12-0-1). In 1972, Miami became the 2nd team to do that.
Happy Holidays :-)
Through the 2007 season, there have been four undefeated/untied teams in the regular season in NFL history:
1. 1934 Chicago Bears: 13-02. 1942 Chicago Bears: 11-03. 1972 Miami Dolphins: 14-0 (Perfect Season)
4. 2007 New England Patriots: 16-0
Of these four teams only the 1972 Miami Dolphins won an NFL Championship/Super Bowl and finished the entire season undefeated (17-0 including playoffs). The other three teams lost the NFL Championship/Super Bowl game.
Before the merger with the NFL, the 1948 Cleveland Browns had the only undefeated season in All-America Football Conference history, finishing the regular season 14-0 and going on to win the AAFC championship game.