Narek was a Romulan spy. He once claimed "Narek" was merely one of his names, and on another occasion that his true name was "Hrai Yan". He had an older sister, Narissa, who was also a Romulan operative. Narissa was surgically altered to resemble a Human for the purpose of infiltrating Starfleet, under the name Lieutenant Rizzo. (PIC: "Maps and Legends", "The Impossible Box")
After a failed attempt to capture Dahj Asha alive, another attempt was orchestrated to go after her twin, Doctor Soji Asha. Narek was assigned for this task and it provided an opportunity for Narek to prove his worth. Not only for the Tal Shiar but in the eyes of his sister as he was considered the "family failure" because he had dropped out of the Zhat Vash, the cabal for which his parents had served and died. (PIC: "Et in Arcadia Ego, Part 2")
In 2399, at the Romulan Reclamation Site where she worked, Narek approached Doctor Asha. He began drawing her in with charming small talk, and admired a necklace she wore, which matched an identical necklace her twin Dahj had worn. He won Soji over, in part, by mentioning a brother he had recently lost. (PIC: "Remembrance")
They quickly began an intimate relationship, part of Narek's assignment in a clandestine operation, in collaboration with Narissa and General Oh, to gather information about the whereabouts of other androids like Asha. (PIC: "Maps and Legends")
Despite his sister warning against it, Narek admitted he'd begun to fall in love with Soji. (PIC: "The End is the Beginning") Narek was criticized by Narissa for falling in love with a synth, and for using a tan zhekran as a thinking aid. (PIC: "The Impossible Box")
He was dispatched on a Snakehead to track La Sirena when it fled from the Artifact. He lost their tail when the tracking device was destroyed. He stayed near a Borg transwarp hub until La Sirena came and went through, at which point he followed it. (PIC: "Broken Pieces")
After his ship was compromised by an orchid, it crashed on the planet Coppelius, where he was eventually captured by the planet's inhabitants. After being framed for the "murder" of Saga by Sutra, he "escaped" and eventually found his sister on the Artifact, which had also crashed on the planet.
On the Borg cube, he gathered a number of molecular solvent grenades and made off for the crashed La Sirena. There, he contacted that ship's crew with the intent of defeating the planet's inhabitants together, using the grenades. He later "allowed" himself to be taken prisoner, which was used to the advantage of La Sirena's crew to allow them into Coppelius Station to allow them access to the beacon that was being built there. (PIC: "Et in Arcadia Ego, Part 2")
Narek was played by Harry Treadaway.
Narek's disappearance from the narrative, after pleading with Soji to disable the transmitter, left his fate unrevealed. According to showrunner Michael Chabon, a scene depicting Narek being taken into Federation custody was ultimately cut from the episode. 
In a blog post on Medium, Chabon elaborated Narek's multiple names were meant to reflect an idea that Romulans are given four different names at birth, which they use depending on how close the person they are speaking to is. "Romulans change their open names at will, often making use of three, four or ﬁve at a time, in different social contexts. Then there is the 'true name,' used only at moments of deep intimacy."