This is the sixth incarnation of Lord Vishnu.
Legend is that Lord Vishnu took this avatar, following a complaint from Bhoomidevi- the mother earth.
At that time the earth is full of arrogant rulers or Kshatriyas (the ruling class) who paid scant attention on preserving dharma on earth. They became a burden to Bhoomidevi; so she approached Lord Vishnu to find a solution.
Thus Lord Vishnu took birth as Lord Parasuram to reduce the weight of mother earth by killing bad rulers. He was born to Sage Jamadagni and Renuka.
He started his mission of eliminating Kshatriyas by killing the mighty ruler Kartavirarjuna. One day, while on hunting, king Karatvira came to Jamadagni’s ashram. The sage served a sumptuous meal to the king with the help of Kamadhenu- a divine cow that Jamadagni up brings at his ashram.
The king got attracted to the divine cow; he wanted to carry it to his palace. Jamdagni rejected his request and in the ensuing tussle, the sage got killed by King Kartavira. This provoked Lord Parasuram ; he beheaded king Kartavira.
Thereafter, to fulfill his mission, Lord Parasuram took 18 rounds on earth and killed bad rulers all around. After his mission was over he met Lord Sriram, while Lord Sriram was way back to Ayodhya after marrying Devi Sita.
Lord Parasuram challenged Lord Sriram to lift the bow that Lord Parasuram carries. When Lord Sriram did so, Lord Parasuram became happy; he gifted all his divine powers to Lord Sriram and went for a long meditation.
But worshiping Lord Parasuram in temples seems rare. One notable exception is Tiruvallam temple in Kerala, near Trivandrum.