Wikipedia's article on Team Fortress is excellent - as usual. A great starting point.
Since the main Team Fortress article did not have a list of classes or rule variations, this is the one I advise people to read first.
If you are too lazy to read that, or what a little synopsis first, here goes. First off, the main format is like Capture the Flag. In fact, it is pretty much as old as Internet based Capture the Flag - it was the first major project to liven up the old CTF formula.
Two teams of players try to take the opponent flag while maintaining their own. For gamers today, this concept is ubiquitous (except in Battlefield, but my bone to pick with them is too large an affair for this entry). What else is different about it?
I cannot do justice to the wikipedia entry, in this area, but suffice to day, class based play is really what gives TF a wealth of depth that CTF does not have. The specialization, and intricate differences between the types of weapons, items, armor levels, and other miscellaneous abilities makes a simple game more complex, more interesting, more strategic, and more fun. If CTF is soccer (or Association/World Football), then Team Fortress is like American Football. You have your linemen, who are totally built for one role in the team, and your quarterbacks, and your receivers. All these different sizes, speeds, and toughness make for a gaming experience where you can focus on particular facets of the teamplay if you wish, or f you don't, there are classes that are pretty good at everything, but great at nothing.
In the coming weeks, I will certainly cover all areas of all classes. But while the class system is the foundation of Team Fortress, it isn't the only thing. The cool variations of grenades, the charging bases sniper system, the variations of victory conditions and flag behavior, and the team oriented online community where all things I miss greatly from Team Fortress that no other first-person shooter I've ever played has surpassed.