I'll try to give you a Cliff's Notes version.
NCAA football is divided into four different divisions, with Division 1-A being the one you hear about all the time. The others are 1-AA, II, and III. They have different rules in regards to scholarships and school sizes. Forget the last three exist for a moment, though, and focus on 1-A.
The schools are split up into conferences, and play a certain number of games within them. Generally, you'll play either all of your conference opponents or at least most of them, and some conferences have 12 or more teams, so they split into divisions of 6. Those conferences have it so you play your division every year, plus a few schools in the same conference, but in the other division. Regardless of the setup, conference play is almost always 8 games. There are also Non-Conference games, in which schools play other schools from other conferences. Most power schools tend to schedule one or two "cupcakes," sometimes even dipping into the 1-AA ranks, in order to get their teams ready to play in conference.
You hear a lot about the BCS, and the "BCS" conferences include the Pac-10, the Big 10, the Big 12, the SEC, the ACC, and the Big East. Their conference winners, plus four more schools known as At-Large schools, which could be 2nd place finishers in the BCS conferences or winners of what are called "Mid-Major" conferences, those that aren't included in the BCS deal under normal circumstances, but have achieved well enough to place above a certain ranking in the BCS standings, which take a few human polls and computer rankings and throw them all together to say who the best teams are. #1 and #2 in those standings play in the National Championship game, and the other 8 schools that qualify for a BCS game play in the four BCS Bowls, the Rose, the Sugar, the Fiesta, and the Orange, and those games are played right around New Year's Day, with the National Championship rotating between the four locations, and played a week later.
The controversy around things is that the computer rankings and the human polls don't always agree, and it can lead to odd results, when looked at from the human perspective. This leads to debates and fights between the fans of the given teams and other college football fans as to who really is better. A playoff system would fix such things, but then you'd have to find a way to select the teams.
I hope this has been a good start, anyway. Ask away if there are things you'd like cleared up.