Kapiraj (the monkey king) Kesari, with his wife Anjana, lived on the mount Sumeru. To get a good son, Anjana worshiped Lord Shiva for years. Lord Shiva was pleased with Anjana's worship and devotion and asked her for a boon. At this Anjani prostrated before Him and humbly requested Lord Shiva, "O, Lord Shiva, I want a strong son, who must have all the virtues."
Lord Shiva said, " So be it. I shall take birth in the form of eleventh Rudra through your sacred womb. Take this Mantra. Vayu, the wind God will give you Prasaad. When you eat that Prasaad you will give birth to a virtous son".
Saying so, Lord Shiva disappeared. Anjana acted upon Lord Shiva's advice and began to Jap the Mantra with full devotion. While she was doing so, a vulture came flying, taking the bowl full of celestial rice pudding in its beak. At that time there arose a heavy wind and hurricane. Due to it, the bowl fell down on the earth. Vayu, the wind god took the bowl and gave it to Anjana. As Lord Shiva had already told her what she was to do. so Anjana took the bowl full of celestial rice pudding and ate it up. She became pregnant and gave birth to Hanuman in due time.
It is believed that Hanuman was born on Tuesday, Chaitra Shukla Purnima. Some others belive that Hanuman was born on Chaitra Shukla Ekadashi in the Magha nakshatra. Some others believe that he was born on Kartik Krishna Chaturdashi and Some believe that his birth Tithi was Kartik Shukla Purnima. Some say that the day was Tuesday and some others say that it was Saturday.
(Source:- Hanuman Ank Page- 249- 250)