1. Historical Conflict Wargames - conflict modeling of historical events.
2. Historical Hypothetical Conflict Wargames - conflict modeling of possible conflicts from history.
Ex. The Battle of Mine Run in the Civil War as done in 1863; Conflicts during the Cold War that did not take place - SPI's The Next War, for example. SPI did a lot of what were 'Future Conflict Games' that were all hypothetical but entirely possible.
One can also say that the 'original' wargame concept is this genre.
3. Science Fiction Conflict Wargames - based in fiction completely - The Star Trek Based games, the book Starship Troopers, the 'Ares' games by SPI.
4. Fantasy Conflict Wargames - Lord of the Rings, etc. Very much Fiction.
5. Abstract Wargames - Chess, Stratego, Battleship, Go, Risk - not based on any historical conflict, hypothetical conflict, or a specific piece of fiction.
Personally, I very much prefer to play Cat 1 Wargames - 90% of my experience is there.
For Cat 2 - less than 5% - I have played many an SPI Hypothetical Historical Game. Beyond that, nothing.
Cat 3 - More than Cat 2 for me - I have enjoyed many a classic SFB game, some Starship Troopers, and I get a kick out of The Creature That Ate Sheboygan.
Cat 4 - I won't waste my time on that genre. I just have a personal dislike of it - though I have done perhaps 2-3 D&D type games a long time ago.
Cat 5. Yeah, I have done Risk, Battleship, Stratego, Chess - all of them pretty much in the 'Sophisticated Family Game' class of boardgames.
The closest I have experience with might be called 'pre-Euro' - games like Junta, Diplomacy, Starforce, Imperium, etc.
Might give one a try if the opportunity arises - but so much wargaming!