Rudra is one of the most popular names of Shiva used extensively in the Vedas to refer to him. In fact, an expression addresses him as Rudra Shiva. The literal meaning of the term Rudra seems to be roaring storm. The other meanings associated with this word are fire and fiery red. Looking from several angles, the term Rudra appears to focus on the fiery aspect of Shiva which is destruction.
Hindu theology talks of three principal manifestations of the Supreme Godhead namely Brahma the creator, Vishnu the protector and Shiva the destroyer.
In this context, the fierce name of Rudra seems to be highly fitting the role of Lord Shiva.
The term Rudra is also associated with the term Rudra Tandava, the terrific dance of Shiva which he performs in the cremation grounds.
The iconography of Shiva shows him wearing snakes as ornaments, a garland of skulls, the ashes of the cremation grounds smeared all over the body and wandering forth in red angry eyes.
A puranic story narrates an incident associated with the term Rudra. Once Brahma asked Rudra to create some beings as he was getting bored by creating ordinary mortals. Owing to this request, Shiva created 11 immortal beings who are called Kapali, Pingala, Bhima, Virupaksa, Vilohita, Ajesha, Shasana, Shasta, Shambhu, Chanda and Dhruva. They were together called as 11 Rudras.
Being the head of these immortal beings, Shiva is addressed as Rudra. The hymns of Rg Veda have a lot of references to Shiva in the name Rudra. They all portray Shiva as the highest God, a mighty archer, the most terrible form, the fire god and so on. In all these expressions, the fiery aspect of Shiva is obvious.
It is a matter for thought that the name Rudra is in sharp contrast to the name Shiva in terms of the meanings suggested by these words. While the term Rudra embodies in it a terrible aspect by denoting the roaring storm, the term Shiva points out to the benign aspect of Shiva.
Thus, these two terms together bring the contrasting features of Shiva put in one expression.
The expression Rudra is extensively used in the Vedic hymn Rudram that is dedicated to singing the praise of Lord Shiva. The summary of this hymn heralds the different aspects of Lord Shiva and praises him as the supreme one who is the one point origin of the entire universe and everything associated with creation. Rudra is also conceived as the final destination of the entire universe in to which it merges back after dissolution.
We can take that while the term Rudra points out to the role of Shiva namely destruction, the term Shiva points out to the benign nature of Shiva which is blessing the devotees with the boons that they want.
Shiva is compassionate and merciful. Being the parent of the entire universe, he is endowed with the supreme role of taking care of their well-being. Chanting the name of Rudra shall invoke the blessings of Shiva and win his grace for the devotees