The National Football League has grown immensely since its conception in 1920. At the time, the league was known as the American Pro Football Association and played its first season with 14 teams, two of which remain (The Chicago Bears and the Arizona Cardinals). It wasn't until 1970 that we saw the merger that created the league as it is today.
After the creation of the American Football League in 1959 by Lamar Hunt, it became apparent that the league was able to compete with the NFL, garnering over a million fans at games by 1962. After multiple disagreements and unspoken rules were broken, the two leagues decided the best option for the both of them would be to merge the two leagues. The agreement was written in 1966 and then became official in 1970 with the league keeping the NFL name.
The problem then arose of splitting the league for the new playoff layout with the addition of eight new teams joining the next season. This time was different than the NFL's previous merger in 1946 in which they only absorbed three of the All-America Football Conference's seven teams.
Thus the National Football Conference and the American Football Conference were introduced, each boasting 13 teams. At first, the new conferences were made up of three divisions on either side with divisions of four, four, and five teams. The NFL quickly switched to the four division system of North, East, West, and South with four teams in each as we know today.
The American Football Conference is divided into four regions of North, East, West, and South.
The NFC's team divisions follow the same blueprint of four regions with four teams in each across the country.
As the fans have noticed, as teams relocate the divisions don't seem to match up with the geographical destination they are named after anymore. However, because this system has worked so well for the NFL they have kept the teams divided as they are to induce the most rivalries.
The NFL's divisions serve to make scheduling and playoff seeding easier, as well as maintain rivalries between division members.
The NFL scheduling system follows a formula that completely depends upon its divisions. Each team plays the three other teams within their division twice, once against each opponent from another division in their conference, twice against the two teams who finished in the same position as them within their respective divisions from the same conference, and once against each team from a rotating division in the other conference.
In the NFL Playoffs, the team with the single best record from each division is given an automatic berth into the playoffs. This system comes under fire a lot for rewarding teams in weaker divisions, while punishing teams in more competitive divisions.
Lastly, the divisional system protects and promotes division rivalries by ensuring two games per regular season against divisional opponents. For example, the NFC East has great rivalries between all four of its members that came about by playing each other frequently for so long.