Answer
According to history, the Arizona Cardinals are the oldest team in the NFL, having originally been founded in 1898 as the Morgan Athletic Club. They eventually became known as the Racine Cardinals and joined the NFL in 1920 as one of the league's founding teams, but changed their name to the Chicago Cardinals a few years after joining the league. They moved to St Louis in 1960 where they played for 27 years before moving to Arizona in 1988.
The Chicago Bears and the Green Bay Packers are the next oldest teams, both having been founded in 1919. However, the Bears were originally founded in Decatur, Illinois, and were known as the Decatur Staleys before moving to Chicago in 1921. The Staleys joined the NFL in 1920 while they were still located in Decatur and, alongside the Cardinals, became one of the league's founding teams. The Packers, however, did not join the league until 1921, and therefore are not considered one of the founding teams of the NFL. Therefore, they are considered the third oldest team. What's worth noting about the Packers is that out of all the teams that joined the NFL within it's first (nearly) 13 years, they're the only team that still has not been renamed or moved to another city. They're also the only non-profit, community-owned major league team in all of professional sports within the United States.
The fourth oldest team is the current New York Giants. They were founded in 1925 and joined the league that same year. Contrary to popular belief, the New York Giants of 1919 and of 1921 are not the same as the Giants of 1925, and have no relation whatsoever to that team except for the name. The Giants of 1919 folded in the same year, and the Giants of 1921 folded two years later, in 1923. New York has had three separate football teams known as the Giants, but only the first two were related.
Finally, the fifth oldest team is the Detroit Lions. They were founded in Ohio in 1929 and were known as the Portsmouth Spartans. They joined the NFL in 1930 and played as the Spartans until 1934, when they were moved to Detroit and renamed the Lions.
If you want to get really technical, the Green Bay Packers could be considered the oldest team, as they can trace their lineage back to 1896 to other semi-professional teams who played in Green Bay. However, they did not officially become the Packers until 1919.