There are number of classical dances and art performance in India, Eight well known classical dances of India recognized by Sangeet Natak Akademi are Bharatanatyam, Kathak, Kuchipudi, Odissi, Kathakali, Sattriya, Manipuri and Mohiniyattam.
Classical dance forms are different then traditional martial dance arts of India, follow the link to know more about Indian martial arts.
Kathak
Kathak is one of the eight forms of Indian classical dance and is found in three distinct forms of gharanas, Jaipur, Banaras, and Lucknow. Kathak dance differs from the Bharatanatyam dance in several ways and is also different from Kathakali, though all three are Indian classical dances.
Bharatanatyam
Bharatanatyam classical dance form originated in Tamil Nadu and is a solo dance performed exclusively by women with religious themes of Shaivism, Vaishnavism, and Shaktism.
Bharata Natyam is traditionally a female solo dance art and there are gestures used in Bharatanatyam, sign language based on hands, eyes, and face, and also contains at least 20 Asanas of Yoga.
Kathakali
Kathakali is most popular classical dance arts of India and a major forms of classical dance of Kerala along with Mohiniyattam. The traditional Kathakali, theory and foundations are same as other classical dances of India mentioned in Natya Shastra.
Kuchipudi
Kuchipudi is also one of the major Indian classical dances and is most closely related to the Bhagavata Mela performance in Tamil Nadu. The dance drama performance arts of Andhra Pradesh is traditionally a night performance, musical instruments like Mridangam and Tanpura are used for Kuchipudi performance.
Odissi
Odissi classical dance from Odisha is performed predominantly by women and again a dance-drama performance art. The Odissi dance originated from the Hindu temples of Odisha, particularly of Jagannath religious stories as well as Shaktism and Lord Shiva and Surya God.
Sattriya
Sattriya's classical dance form originated in the state of Assam and combines the themes of Krishna and Radha, sometimes Rama and Sita. The costume of the Sattriya dance is unique to this dance form along with jewelry and styles.
Manipuri
Manipuri or Jagoi dance is particularly known for its Radha-Krishna called Raslila themes. The Manipuri dance is also based on religious themes related to Shaivism, Shaktism, and regional deities.
Mohiniyattam
Mohiniyattam is one of the most popular classical dance forms from Kerala along with Kathakali. The solo dance has roots in the Natya Shastra and some of the sequences of Mohiniyattam are similar to Bharatanatyam.
Yakshagana
Yakshagana is a unique style and art form, developed in the state of Karnataka. Classical music and theater forms are broadly classified into several types such as Tenkutittu and Badagutittu.
Chhau Dance
Chhau dance is actually semi classical Indian dance that origins in eastern India and are performed in West Bengal, Jharkhand, and Odisha. The dance is mixture of martial arts, tribal forms and folk traditions of India, Chhau mask of Purulia is registered under Geographical Indications.
Bhagavata Mela
Bhagavata Mela dance performed particularly the Thanjavur area of Tamil Nadu and the dance art has roots from Kuchipudi classical dance of Andhra Pradesh.
Gaudiya Nritya
The Gaudiya Nritya is a classical Bengali Dance form ,performed with drama, history, poetry, color and music. Its an ancient classical dance from West Bengal, Mainly a temple art meant for spiritual expression.