The school claims 14, the NCAA actually recognizes us as having 18 National Championships. They say 18 because there are other organizations (committees, magazines, etc.) that were around in the past that the NCAA recognized as having "official" National Champions. The main reason that ESPN and other douchebags want to dispute 5 of our titles is because they were pre-AP Poll (before 1936)... the main argument here is that almost every year multiple teams were named National Champions, primarily because #1 rarely EVER played #2 in a bowl game and alot of times National Champions were selected BEFORE the bowl games (even in the Poll era that happened). There were so many shared championships in those days is the main argument.
The 1945, 1962, 1966, 1975, and 1977 teams are recognized by the NCAA as championships for Alabama. In '45 Alabama was undefeated and beat the hell out of USC in the Rose Bowl, but Army was also undefeated and #1 pretty much the whole year so they were given the "official" National Championship and AP National Championship. The National Championship Foundation picked Alabama as the National Champions tho and the NCAA recognized it but Alabama doesn't.
'62 Alabama was 11-1 and despite USC almost being unanimously selected champion Billingsley selected Alabama. Alabama lost to an unranked Georgia Tech that year, USC went undefeated.
'66 is one of the more controversial ones in NCAA history (there is a really good book called "The Missing Ring" you should check out), Alabama was 2-time repeating National Champions and was looking for the near impossible 3-peat. They went undefeated the entire year destroying almost everyone they played including 6 shutouts, and 4 in a row. Despite being unquestionably the most dominating team in the country, and obliterating the much bigger Nebraska in the Sugar Bowl they were voted #3 at the end of the year. Now, there is and was ALOT of controversy over this for many reasons. 1) #1 Notre Dame played #2 Michigan State during the season and with the game tied late in the game Notre Dame's coach played for the tie, running the clock out instead of trying to win the game (could you imagine Bear Bryant even THINKING about that?) not wanting to risk turning the ball over and possibly losing. Despite that Alabama NEVER jumped either team and neither Notre Dame or Michigan State played in a bowl game (thats part of back in the day college football i still dont understand). The AP & Coaches Poll gave Notre Dame the National Title (OVERWHELMINGLY so) and one other entity gave MSU their share. The Berryman Foundation is the only gentility to give Bama the championship but pretty much noone honors it (other than the NCAA itself) so neither does Alabama. Now, the biggest part of controversy is the theory of WHY Alabama wasn't given the respect it deserved. Primarily because of the race factor. This was smack dab in the middle of the civil rights movement and Alabama remained all-white and many saw its dominance in football as proof of the white race being superior (ignorantly on both sides of the argument) and many people in the media (and in the public outside of Alabama) sat the blame right at Bear Bryant's feet as to why Alabama's team wasn't integrated yet (now Bryant had his reasons but it can be argued that he should have acted sooner but thats another argument for another time). Some felt it was the media's way of "sending a message" to Bryant and Alabama.
In 75 Alabama was 11-1 (lost the opener to Missouri and then won every single game of the year by double digits until the Bowl Game which it won by 6). There was 4 teams given a share of a National Championship that year (Alabama, Arizona Sate, Oklahoma, & Ohio State) and Alabama was the only one to only get a national championship from one entity (Matthews) and they dont claim it.
In 77 Alabama again 11-1 (and beat #1 USC during the year), #5 Notre Dame (which lost to an unraked Ole Miss that was beaten by 20+ by Alabama) beat #1 Texas (beat the dog sh*t out of them) in the Cotton Bowl and jumped #2 Alabama to win the National Championship. The Football Research Foundation gave Alabama the National Championship but Arkansas, Notre Dame (almost unanimous), and Texas (yes even after getting monkey stomped in the bowl game) were also given shares.
I know I rambled some but I've done an essay on this and on the '66 team (and on race in college football).