Turing is acquired only in the OMNI Mainframe mission, after Monst3r has completed hacking the Security Hub. When Turing is acquired, if you do not have a spare Incognita program slot then a new program slot is created for it.
The program Turing is named after the real-life computer scientist Alan Turing. Alan Turing was a pioneer of early computer science, famous for his work on cracking the Enigma cipher during World War II, and his proof that the Halting problem has no computational solution. These are referenced in the game by the Turing program icon, which is a stylized representation of an Enigma machine rotor, and Incognita's voice line “computation halted” when the program Turing is used.
The program Turing is given to the player relatively late in the mission, after the first objective has been completed. At this point in the mission, Monst3r needs to make his way from the Security Hub to the Mainframe Lock; Central needs to get there too, if she isn't already.
By this time, many of the devices in the facility which you want to hack have already been hacked, but some have been recaptured by Failsafe. Turing can be a useful way to hack any recaptured Security Cameras, lower the remaining armor of OMNI Protectors, and break some firewalls on the Mainframe Lock all at once. If you need to do several of those, then Turing can be a more efficient use of PWR than your other programs; if not, then you may not need to use it at all.
Turing breaks firewalls on all devices, including those with daemons installed. If there are devices on 1 or 2 firewalls with severe daemons such as Validate, Labyrinth or Modulate, then it is usually best not to use Turing. On the other hand, if there are no particularly harmful daemons installed on any such devices, then triggering weak or ineffectual daemons can be a beneficial side-effect of Turing due to the PWR bonus from Central's Antiviral Proxy augment.