Rudra is evolved from the combination of Rud (in Sanskrit means Sin) and Dravayati (means washed away). Thus the process involved or to say the Shanti performed to wash away our sins is called Rudra.
The underlying principle is that we, human beings, knowingly or unknowingly commit sin in our day-to-day life. There are various types of sin we commit, for say, verbally, (abusing or being harsh on someone), through our thoughts (ill feeling towards someone) and physically (forcefully).
We need to apologies towards this and as such Rudra is performed. There are several types of Rudra of which Rudraikadashani (Laghurudra swahaakaara yagna)/ Maharudra swaahaakaar yagna) are generally performed to purify ourselves by apologizing for our sins.
***Sri Kalahasti is the Vayu Kshetra (Wind Representation) among the five places dedicated to nature. It is situated in Chittoor District, in Andhra Pradesh. The light in the sanctum sanctorum is always seen gently swaying as if blown by a wind. The blowing is caused; it is said, by the breath of Lord Shiva. The lingam was worshipped by a Spider (Sri), snake (Kalash), and an elephant (Hasti), hence the name as Sri Kalahasti. Maha Shivaratri is celebrated with great reverence and fervor in the temple.
The following information on the Rudra Yajna has been shared by Brother Kumar Mahadevan
The Most Wonderful "Rudra Yajnam" is to be performed in Prasanthi Nilayam next month. Here are some facts about the Yagnam; it has been performed only once before by Shiva for the Good of the World, Now it is performed by Sai Shiva.
The Rudra Yajna is the Most sacred and powerful among all the Maha Yajna's encompassing two Nishkama Karmas, namely, Deiva Yagna (worship of Shiva and Other divinities), and Manushya Yajna (performed with Large devotees who partake prasadam or grace of the Lord). It is very rare that a Yagnam of this significance and magnitude can be witnessed. Equally rare is the opportunity for the people to actually participate in the worship. This most powerful Yajna is done only with the Presence of a Divine Being and done in a most serene environment for meditation, spiritual study and practice. It is also done particularly for the sake of washing away ones sins and afflictions, for bringing peace, prosperity and happiness, along with family togetherness.
The Sri Rudram chant is an ancient Vedic hymn, and is the oldest example of a listing of various names of Lord Siva. Through the chanting of Sri Rudram, Lord Siva's various attributes and aspects are invoked and worshipped. Chanting the Rudram is considered to be of great benefit. The Rudram chanting can be done with or without the accompaniment of a Vedic yagna ritual. When accompanied with the Vedic fire ritual, it is called the Rudra Yagnam.
It is said that Lord Shiva after Basmasura was killed with the Help of Lord Maha Vishnu performed the Tandava Dance and then performed the "Rudra Yagna" for the Betterment of Humanity, this Place where the Rudra Yagna was performed is where the "Sri Kalahasti" Temple stands now, and this temple also has one of the 12 Jyothirlingas of Parameshwara.
There is no other mention of this Rudra Yagna ever been performed after that as this Maha Yagnam has to be performed in the Presence of Divinity itself, Our Sai Shiva is performing it now. That is the significance of this Rudra Yagnam and we are about to witness something Divine and which has happened only once before.