Abstergo Industries is the fictional company that controls the Animus in Assassin's Creed's present-day storyline.
Abstergo is a modern multinational corporation secretly controlled by the Templars. Its public operations include pharmaceutical research and other corporate activities, while its hidden agenda involves locating Pieces of Eden and using genetic memories to investigate the past. The Animus is the technology that lets a user relive ancestral memories encoded in DNA.
In the first Assassin's Creed, Desmond Miles is taken to an Abstergo facility and forced to use the Animus to experience Altaïr's memories. Later games show that Abstergo has commercialized Animus technology through entertainment products, allowing the company to shape how historical experiences are distributed.
Abstergo is not the same as the Templar Order itself. It is the modern corporate organization through which the Templars exert influence. The Assassin Brotherhood is its principal ideological opponent, although the modern conflict includes many allies, splinter groups, and competing institutions.