A Puranic story explains how Shiva chose to manifest Himself as Nataraja, the Lord of dance at Chidambaram.
The Rishis of yore residing in the Daruka forest were great advocates of the karma marga. They were given to performing elaborate yagnas as prescribed in the Vedas and believed that such karmas would by themselves lead to the highest Liberation.
Shiva, in order to free them of their delusion descended upon Darukavana (where the Rishis were performing a huge yagna) along with Bhagavan Vishnu disguised as Mohini.
The Rishis were captivated by the beauty of Mohini and went after her. The wives of the Rishis on beholding the Bhikshatana form of Shiva were fascinated and followed Him.
The Rishis were enraged and they realised that the sanctity of their yagna was being vitiated. Vowing to punish the one responsible for this they began fighting with Bhikshatana without realizing it was Shiva.
They used all kinds of weapons and finding that none worked against Shiva they started producing objects from their sacrificial fire. Shiva seized each of those weapons and made them his ornaments.
The bewildered Rishis surrendered to Shiva and realized His true identity.
Shiva then danced the Tandava for them and expounded the doctrine of Brahman, the Supreme Self.
After returning to Vaikunta, Bhagavan Vishnu narrated to Adisesha, the king of the Serpents, the story of the dance of Shiva. On hearing the exploits of Shiva, Adisesha developed an intense desire to watch His dance and to this end performed rigorous tapas.
Shiva appeared and promised Adisesha that he would dance at the Tillai forest situated in the southern part of Bharatavarsha on the afternoon of Thai Poosam day which would fall on a Thursday. Adisesha then decided to proceed straight to Tillai. He exhibited extreme reverence and devotion by descending on earth with his palms joined in obeisance. He was therefore known as Patanjali, the one who came with his palms joined in namaskara.
Patanjali appeared at Tillai, the upper part of his body resembling a human one and the lower part resembling that of a serpent. Vyaghrapada, another devotee of the Lord, who performed intense tapas with the intention of witnessing His dance, was also directed to Tillai. Vyaghrapada, in order that he may be able to climb trees swiftly and pluck bilva leaves and flowers for Shiva puja, prayed to Shiva that he may be blessed with the feet of a tiger. Shiva granted his request and thus it was that he acquired the name Vyaghrapada.
It was the exemplary devotion shown by Patanjali and Vyaghrapada that persuaded Shiva to manifest as Nataraja in the Tillai forest and to perform the Ananda Tandava to the delight and benefit of the entire universe. On the appointed day as promised to Patanjali and Vyaghrapada, Shiva danced at Thillai amidst Brahma, Vishnu, Saraswati, Narada and Nandikeshwara who played musical instruments, and the celestials who watched awe-struck. In response to the prayers of all those assembled Shiva agreed to make the Tillai forest his permanent abode and perform the Tandava dance, the true significance of which is the vision of the Supreme Self dancing in the heart-lotus of the devotee.
Source – excerpts from Nataraja: The Lord of Dance By M. Ravindra Narayanan published in the The Mountain Path Aradhana Issue 1998 page 23 - 24