Until 1968, these were two separate and unrelated leagues. National Football Conference and American Football Conference are two different leagues which came together as NFL on popular demanded. This is much like the Major League Baseball which consists of the American League and the National League although the rules across the NFL are the same as opposed to the difference in the MLB where AL has the designated hitter while NL does not allow it.
There is no real difference in the teams and both leagues have 16 teams each. However, there is a difference in the playoffs and winners in a given league receive a different trophy. A number of teams play a part in the postseason and give it their best shot to win their conference silverware. The teams eventually play the big final, known as the Super Bowl to determine the champion of both leagues or simply the NFL champion.
The teams have to compete for the trophy of their own league and therefore do not play in both leagues at the same time. Once a team is the part of the AFC or the NFC for the given season, it must stick with that league and cannot be playing a part of the other league at the same time. Otherwise there will be an issue of judging the team playing in two different leagues which may not be possible and will cause confusion.
The teams are divided into groups depending on their geographical location. The teams are divided into North, South, East and West Divisions.