He is born in a Muggle orphanage to a witch mother, Merope Gaunt, and Muggle father, Tom Riddle. His mother Merope dies an hour later, living just long enough to name her son after his father, Tom, and her father, Marvolo Gaunt (HBP13). She is nineteen years old. The older Tom Riddle had previously left Merope when he realized that she was a witch (HBP10), and the child is raised in the orphanage (HBP13).
New Year's Eve is given in HBP13. Depending on exactly when Merope Gaunt arrived at the orphanage, Tom may have been born on New Year's Eve or just after midnight, on New Year's Day. Rowling was asked on her website to clarify the actual day:
Q: We learned in book six that Merope Gaunt staggered into the orphanage of New Year’s Eve and gave birth to Tom Riddle ‘within the hour.' Was Voldemort born on December 31st or January 1st?
A: He was born on December 31st (JKR).