There are 32 teams. There are two divisions, the AFC and NFC which both contain 16 teams. Each Division has 4 Conferences with 4 teams in each conference. Each team plays 16 games during the "Regular Season" which consist of playing each team in their own division twice (home and away), each team in the opposite division in the other conference once (Teams in the NFC North play the teams in the AFC North). The rest of the games are random. At the end of the "Regular Season" we start a knockout round called the Playoffs. The Playoff teams consist of the division winners and the two teams in each conference with the best record called the Wild Card teams. So there are 4 division winners and two Wild Card teams in the playoffs of each conference.
The two wild card teams play away at the two worst division winners. The two best teams get a "bye" into the second round. The winners of the first round meet the two best teams in the second round and the Best team always gets to play the worst winner from round one. Then the winners of those games play each other and as usual the best team gets to play the game at home.
Once you have the Conference Champions from both the AFC and NFC they play at a neutral site in the Super Bowl for the League Championship.
Remember the Titans is about a high school team. In the NFL because of the playoffs each team can afford to lose some games in the Regular Season, but High School and College are different. High School works much like the NFL, but there are far fewer regular season games, and many more playoff games where the teams compete for the State Title in their own states. For example, a team would compete for the Florida State Title.
In College it is generally accepted that you need to go undefeated in order to play for a National Championship. The College format can be extremely confusing, so I'll try to keep it simple. Since there are about 120 teams in college football and no playoffs, it is difficult to figure out who will win the National Championship. To determine who is the best team, the Coaches and also the Media and a number of other sources release rankings every week. For example this week Ohio State is the #1 Team in the Country.
At the end of the season they take a complicated formula that factors in numerous polls, strength of schedule, and number of losses to determine which two teams will play for the National Championship.
American sports are different from European Sports in that we generally combine the Cup and the League into the same thing. We have the League at the beginning to determine seeding for the Knockout Round. If you follow basketball you should be somewhat familiar with this system already and if you don't, good luck trying to understand it.
College Football is so popular because there's this thing in America where for some reason the general public feels that amature players playing for only the love of the game and the pride of their school is more pure than professional sports. I don't believe this, but I still follow it a ton. First of all, the best players go on to play professionally; in fact you MUST play in college to play in the NFL, and you can't play professionally until you are 3 years removed from high school. This isn't like every other country where you can sign a professional contract at any age.
Oh, and I just graduated from the University of Wisconsin which has a good football team, so obviously I care a little bit about my college team. Same with College Basketball.
In order to paint a picture, imagine Wayne Rooney, Cristiano Ronaldo, and Lukas Podolski playing as amatures for some college 20 miles from your house. It's pretty cool to see those guys in a setting where they aren't really much different from you; I had athletes in my classes and played against some of the scholarship basketball players at the gym. Imagine going to a field one day and Rooney's just kicking around trying to get 22 random people to play a game, that's kind of what it's like in college.