The character of Peter Maximoff / Quicksilver is an adaptation of Comic book character Pietro Maximoff. From the Marvel comics storyline, Quicksilver is the son of Magneto (Eric Lehnserr) and Magda. He is the twin brother of Scarlett Witch.
There are lot of differences between the Marvel comics X-Men storyline and the X-Men movies.
In the book, X-Factor Annual #4 Magneto and Magda escape from concentration camp and got married and settled down (as per the comic book storyline Magneto's name was Max and he changed it later to Eric). After marriage when Magda saw the powers of Magneto she fled out of fear and gave birth to twins Pietro and Wanda. Both were raised as Quicksilver and Scarlett Witch.
P.s. during the movie scene where Magneto attacks the White House and starts the Live TV coverage, Peter (or Pietro a.k.a Quicksilver) is shown watching it on TV with his sister who has red hair, it MIGHT be reference to Wanda (i.e. Scarlett Witch), his mutant sister.