The classical dances:
Kathakali and Mohini Attam from Kerala.
Kathakali literally means story-play and is an elaborate
dance depicting the victory of truth over falsehood.

Bharata Natyam from Tamil Nadu.
Bharata Natyam dance has been handed down through the centuries by dance teachers (or gurus) called nattuwanars and the temple dancers, called devadasis.
A typical nritya pose of Bharata Natyam dance

Kuchipudi from Andhra Pradesh Raja and Radha Reddy in a striking pose of Kuchipudi dance.The dance drama that stil exists today and can most closely be associated with the Sanskrit theatrical tradition is Kuchipudi which is also known as Bhagavata Mela Natakam. The actors sing and dance, and the style is a blend of folk and classical. Arguably this is why this technique has greater freedom and fluidity than other dance styles.
Odissi from Orissa
Gorgeous Odissi pose. Picture courtesy of the Odissi Kala Kendra.Odissi is based on the popular devotion to Lord Krishna and the verses of the Sanskrit play Geet Govinda are used to depict the love and devotion to God. The Odissi dancers use their head, bust and torso in soft flowing movements to express specific moods and emotions
Kathak from Uttar Pradesh
The legendary exponent of Kathak, Birju Maharaj. This north Indian dance form is inextricably bound with classical Hindustani music, and the rhythmic nimbleness of the feet is accompanied by the table or pakhawaj. Traditionally the stories were of Radha and Krishna, in the Natwari style (as it was then called) but the Moghul invasion of North India had a serious impact on the dance. The dance was taken to Muslim courts and thus it became more entertaining and less religious in content. More emphasis was laid on nritta, the pure dance aspect and less on abhinaya (expression and emotion).
Manipuri from Manipur
Singhajit Singh and Charu Siya Mathur.This dance style was originally called jogai which means circular movement. In ancient texts it has been compared to the movement of the planets around the sun.
Folk dance of Kashmir: Rouff
The dance form of Mohiniattam was nurtured in the region of Kerala in southwestern India. The name Mohiniattam literally means 'Dance of the Enchantress,' and it does have a mesmerizing quality. The white and gold costume, arresting hairstyle and the highly graceful movements in medium tempo, contribute to this aesthetic effect.
Modern Dance in India has a relatively short history. Since the perception of 'modern' or 'contemporary' can vary from dancer to dancer, this dance form cannot be defined as easily as the classical dance styles of India. It is also not codified in a detailed manner, as are the classical styles. Uday Shankar, who was born in the early years of the 20th century, is widely accepted as the Father of Modern Dance in India. This great dancer had a very wide vision, and he appreciated the wonderful variety and scope of expression afforded by the different classical and folk dances extant in the country.
CHENNAI DANCE & MUSIC FESTIVAL
Mid December to mid January   Chennai music and dance festival is a celebration of classical music and dance of South India (Carnatic Music) held during mid December to mid January in the capital city of Chennai. The festival is held at a number of venues around the city by various sabhas or organizations.  
KHAJURAHO DANCE FESTIVAL
Khajuraho, Bundalkhand, Madhya Pradesh Date : 25th February to 2nd March (for the next Seven Years from 2002)The small village of Khajuraho situated in the northernmost part of Madhya Pradesh, lies 395 Km southeast of Agra. Once the religious capital of Chandela dynasty, one of the powerful Rajput dynasties of Central India, Khajuraho is now famous for it's enchanting temples and it's legendary Khajuraho dance festival. The week- long festival of classical dances is held every year in February/March against the spectacular backdrop of the magnificently lit temples.
India is very rich with traditional, centuries  old  arts and culture. There is so much to explore. Few are noted below: