Classical dances in India have their origin in temples. They began as a form of worship.
Kathakali is a Kerala dance. It is a pantomime depicting a scene from Mahabharat, an epic, whch Hindus consider as 'smriti'(tradition). It is only performed by men.
Bharat Natyam is of Tamil Nadu origin. It was also originally performed in temples as a form of worship. This form of dance is sensuous and has foot work facial expressions (bhava), delicate body movements(mudra)and acting (abhinaya).
Kuchipudi originated in a village called Kuchipudi, in Guntur district of Andhra Pradesh. Originally it was performed by boys. When boys impersonate girls, they tend to exagerate the feminine gestures. That is how this dance has exagerated faminine gestures. This dance was patronised by Quli Qutub Shah, the founder king of Qutubshahi dynasty.
Orissi originated in Orissa. This dance was formed on the basis of scupted temple figures of Orissa. Since the movements in sculptures were restrited, in this dance foot and hand movements are closer to the dancers' body. In Bharatnatyam there is no restriction of movements.
Manipuri dance is Krishna Lila. This is a costume dance. Since the costume is very elaborate, the vibrancy is affected. However costumes, bhava and mudra are a treat to watch.
Kathak also originated in temples. Because of the patronage of Wajid Ali Shah of Avadh, it has become a secular dance. It has beautiful foot work, bhava, abhinaya and mudra. Most of the Hindi film dances are based on this form.