A Halo Purist is a Halo fan that refuses to accept any changes made to the game. They believe that every game should be exactly like Halo CE, Halo 2, and/or Halo 3.
A perfect example is Halo 4. The purists are throwing a hissy fit because Halo 4 is adding a bunch of new features. They accuse Halo of turning into Call of Duty, just because it is adding some things that COD has.
If Halo was like Call of Duty, then Halo would never change. Microsoft would release a new Halo every year with minimal changes to the game. So, these purists are asking for rehashes, not change. The purists basically want Halo to go the same route Call of Duty went.
Halo Purist : Halo 4 is gonna be COD Space Warfare. Why can't 343i make it like Halo 3?
Halo fan : Because most Halo fans don't want Halo 3.5 unlike you. If you want Halo 3 so bad, go play Halo 3. This is Halo 4, and they're actually making changes to the game.