I'm not sure if you think there are only 2 types (there are allot more), or if you want to know of 2 types.
I will answer the second version, 2 types of role-playing games are real-life table-top RPGs (like dungeons and dragons) and eastern style video game RPGs (like Final Fantasy)
If you wanted to know what types of role-playing games there are, let me know and I will gladly answer.
The Metabarons Roleplaying Game happened in 2001.
Pathfinder Roleplaying Game happened in 2009-08.
Marvel Universe Roleplaying Game happened in 2003.
DC Universe Roleplaying Game happened in 1999.
The Lord of the Rings Roleplaying Game happened in 2002.
Amber Diceless Roleplaying Game happened in 1991.
Star Trek Roleplaying Game - Decipher - happened in 2002.
Star Wars Roleplaying Game - Wizards of the Coast - happened on 2007-06-05.
Roleplaying in the game World of Warcraft
I have played WOW for a long time. I think it is a perfect roleplaying game which make us feel satisfy.
Infamous definitly.
Grendel's Revenge