Kylo Ren knows that Anakin had turned back to the Light. What he knows about Anakin's earlier fall to the Dark Side is unknown as of this time.
Kylo Ren is aware, at the time of The Force Awakens, that Vader had ultimately rejected the Dark Side in favour of the Light — Kylo chooses to view this in negative terms, believing that agents of the Light Side had "seduced" Vader.
In the Alan Dean Foster novelization, Kylo and Supreme Leader Snoke have a conversation in which Snoke says:
"Such a foolish error of judgement. A momentary lapse in an otherwise exemplary life. Had Lord Vader not succumbed to emotion at the crucial moment — had the father killed the son — the Empire would have prevailed."
So, if Kylo hadn't known earlier, then he certainly would have known from this point forward that Vader had rejected the path of the Dark Side, at that he did so at least partly because of an emotional response to Luke.
However, Kylo most likely knew about Vader's decision for some time. This is made clear by J.J. Abrams' own explanation of Kylo Ren's motivations:
Kylo Ren idolizes Darth Vader, not Anakin Skywalker. He idolizes what Vader represents and what Vader was trying to do. And the idea that Vader didn't succeed, if you look at it from Ren’s point of view, he was seduced by the enemy and failed because of that seduction. So the idea is that Ren wants to complete the thing that Vader started.
Abrams' phrasing makes it clear Kylo Ren's own motivations for following the Dark Side have to do with Vader's rejection of it. Kylo sees this as a wrongdoing committed by "the enemy": Luke and other agents of the Light.
The fact that Kylo Ren is aware of this means that someone — at least one of Luke, Leia, Han, or Snoke — had told him during his upbringing or training.
Finally, what Kylo knows about Anakin's initial fall to the Dark Side — his relationship with Padme, his dealings with Palpatine, his fight with Obi-Wan — is unknown at this time. Nothing is said about this in the film or novelization, and no official statements have been issued.