The two factions in assassins creed are entirely fictitious. The ‘templars’ are incredibly loosely based on the Knights Templar, otherwise known as Poor Fellow-Soldiers of Christ and of the Temple of Solomon, a Christian religious order created to protect and help pilgrims, as well as act as a sort of bank on their journey to the holy lands during the crusades. The knights Templar were founded in 1119 and were active until about 1312.

I can happily provide you some sources and further information on those, but the history and background of these two factions based on a game is entirely different for the templars and non existent for the assassins. Assassins worked for a state or political heads, or as mercenaries, but carried out espionage, subterfuge and assassinations as low key as possible. Not like in the games where you are superhuman with advanced technology hundreds of years superior to the time and certainly not part of some gang/cult/faction.

The nearest parallel I can think of are a extreme Muslim Nizari Ismailis, a heretical sect of Shiites more commonly known in the West as the Assassins. The Great Khan Mongke finally put an end to them in 1256.

As a side note; the followers of Ismailis were trained and seduced with extensive use of narcotics, especially hashish. Because of the importance of narcotics for the Ismailis, the people around them called them hashshashin, meaning “the hashish users.” Over time, in the West, this name became modified into the word ‘assassin’.

Sources:

Asbridge, T. The Crusades. (Simon & Schuster Ltd, 2012).

Maalouf, A. The Crusades Through Arab Eyes. (Schocken, 1989).