This quest is acquired from Dr. Zimmer in the Rivet City Science Lab. He wants you to find his missing andoid and sends you to speak with Dr. Preston (also in Rivet) who gives you a holotape. You will either need to collect 2-3 more of these from various place (see the link below), speak with Winthrop in the Underworld, speak with Moira in Megaton or continue doing other things until you are approached by a woman named Victoria Watts (she can show up anywhere). If you collect the tapes or speak with Winthrop/Moira, you'll find that a man named Pinkerton did a surgery on the android. You will find Pinkerton in the broken bow section of Rivet City. He will tell you who the andoid is (turns out to be Harkness, the dude who first greeted you to the city) and you can either convince the man he is an andoid (he'll kill Zimmer), or unlock his memories (Zimmer will reclaim him). If you met with Victoria Watts, she gives you a piece of the android, which you can present to Zimmer as if the andoid is dead which ends the quest much more quickly.
See The Replicated Man for more specific info.