Kalbhairav is an incarnation of Lord Shiva and a complete replica of Shiva. Once Lord Brahma and Lord Vishnu quarrel about their greatness.
The origin of Bhairava can be traced to the conversation between Brahma and Vishnu recounted in the Shiv Mahapuran where Vishnu asked Brahma who is the supreme creator of the Universe.
Arrogantly, Brahma told Vishnu to worship him, he being the supreme creator. One day Brahma thought, "I have five heads, Shiva also has five heads. I can do everything that Shiva does and therefore I am Shiva" Brahma had become a little egoistic.
Not only had he became egoistic, he started to forge the work of Shiva. Brahma started interfering in what Shiva was supposed to do.
At that moment a large Agnilinga appeared beween them. Both the gods tried to see the ends of linga top and bottom respectively. But they failed to reach the end of Shiva linga.
But Lord Brahma lied that he saw the end and Lord Vishnu accepted his failure honestly. On that occasion Lord Shiva came and warned Brahma to stop lying. But Brahma insulted Lord Shiva.
Then Mahadeva (Shiva) threw a small nail from His finger, which assumed the form of Kala Bhairava, and casually went to cut the head of Brahma.
Kal Bhairav cut the fifth head of Brahma. Then Brahma recognized his mistake, and praised and asked Shiva to pardon him. Then Kal Bhairav left Brahma.
Lord Brahma and Lord Vishnu regained their knowledge and left their pride and ego. Then Lord Shiva told that "Ones knowledge will not be glorified until he has pride and ego.
One who removes his pride and ego will know the God. God will destroy persons with extremely proud. The god who removes pride was born."
This means Kal Bhairav will remove Pride and ego, he will not tolerate lying, he loves truth.
In the form of the Kaala Bhairava, Lord Shiva is said to be guarding each of these Shaktipeeths. Each Shaktipeeth temple is accompanied by a temple dedicated to Bhairava.
Other names of Kala Bhairav :
Kala bhairav was also called as Kalaraja as he was looking like Kala ie. Yama.
He was Called as Bhairav as one will get fear when he saw Kalbhairav.
He punish bad persons, hence the name Amardaka.
He removes sins of his devotees immediately, hence the name Papabhakshana.
Characters of Kal Bhairav:
He used Brahma kapala as his Bhiksha patra. He will practice Kapala vrata and bhikshatana (begging) to remove his Brahmahatya dosha.
A lady named, Brahmahatya will follow him every where except in Varanasi (kashi). She had no entrance in Varanasi (kashi).
Kal Bhairav will free from Brahmahatya, when he entered in Varanasi (Kasi). The place where Kal bhairav freed from his Brahmahatya was called as Kapalamochana teertha.