This is The Magic Sword from 1962. Description from Wikipedia:
George (Gary Lockwood) is the foster son of Sybil (Estelle Winwood), an elderly sorceress. She brought him up after his "royal parents died of the plague" in his infancy. He has fallen in love at first sight with Princess Helene (Anne Helm).
She is kidnapped by the wizard Lodac (Basil Rathbone), who brazenly informs her father that he intends to feed her to his pet dragon in seven days, revenge for the death of his sister at the same age as Helene is now: 18. George wants to go on a quest to liberate his lady love, but Sybil believes he is too young (he is 20). She tries to distract the youth by showing him a magic sword, a steed, an invulnerable suit of armour, and six magically frozen knights he will command when he turns 21. The impatient George, however, tricks Sybil and locks her in a cellar, then leaves with the magical implements and revived company of knights.
[omitted are paragraphs of the quest and the knights dying off one by one]
George sneaks into Lodac's castle and rescues Helene, only to be captured. The magician gives Helene (actually the hag in disguise) to Branton, but once he has the ring, he uses magic to put Branton's head on a plaque on the wall. George is tied up, but escapes with the help of shrunken prisoners. Sybil arrives and finally remembers the spell that restores George's powers, enabling him to slay the two-headed dragon and save Helene. Sybil steals the ring while Lodac is distracted. When the magician threatens the young couple with the seventh curse - himself - Sybil transforms herself into a large panther and kills him. The movie ends as Helene and George get married. When the six knights are returned alive, George's happiness is complete.
There is a two-headed dragon, but the two headed monster is actually a set of conjoined twins looking through the magic mirror, as seen in the trailer.
The trailer (and it's not hard to find the full movie online either):
Search terms used: 1950s film knight on a quest to save a princess