6. Well, first, there are the Servant Classes, which, as implied, are used to categorize each of the Servants by their specialty. Some Servants have a varied enough history that they can be applied to more than one Class, while others are limited to just one.
The seven basic Classes that are first introduced in Fate/stay night and have continued to appear in the series are Saber, Lancer, Archer, Rider, Caster, Assassin, and Berserker. The majority of Servants are among these Classes, including most of the "main" Servants over the course of the franchise (the classic Saber from Fate/stay night and Fate/Zero, Red Saber from Fate/Extra, Tamamo-no-Mae from Fate/Extra, the various Servants of the Red and Black factions in Fate/Apocrypha, etc.)
There are also Classes that are collectively referred to as the "Extra" Classes, which do not fall into the seven main Classes. These Classes include Avenger, Ruler, Alter Ego, Foreigner, Moon Cancer, and Saver.
I'd go on, but then this post would just be an even bigger infodump than necessary.