The Animus user controlling Altaïr in the original Assassin's Creed is Desmond Miles.
Desmond is a modern-day descendant of Altaïr Ibn-La'Ahad. Abstergo Industries captures him and places him in the Animus, a machine that allows a user to experience genetic memories inherited from ancestors. Through Desmond, the game shifts between the modern storyline and Altaïr's life in 1191.
Altaïr is therefore not physically travelling through time. The historical sequences are reconstructed memories accessed through Desmond's genetic lineage. This framing device became one of the defining features of the early Assassin's Creed games.
William Miles is Desmond's father and an Assassin leader, while Clay Kaczmarek is another important Animus subject introduced in later modern-day material. Neither is the user controlling Altaïr in the first game.