What is significance of dance in Hinduism?
I think that there can be many different aspects to this answer. One significance would be that the concept of dancing is something strongly associated with the gods. For example, Lord Krishna and his beloved dance the dance of love in their sacred Vrindavan gardens. Their dance is considered to be the very personification of love divine and within their movements a great significance is attached. There can be other dances that might not be so amorous bound. For example, there is Lord Shiva. There are many examples of how Lord Shiva, often personified as the dancing Lord Nataraja, is capable of dancing a furious dance that brings about massive amounts of destruction. An example of this would be when Shiva's first wife, Sati, throws immolates herself as a statement against the disrespect of her beloved husband. Shiva's dance of destruction was one that brought about destruction to all and represented a tantric dance that genuinely scared many of the gods, who had to plead to Lord Vishnu for a way to figure out how to stop this dance of destruction. The Goddess Kali does a dance of fierce destruction in vanquishing her enemies, as well. Dancing can be a way to bring about a sense of destruction to the world when the gods have not been sufficiently pleased.
Dance as well as music has a prominent place in Hindu religious rituals and practices. Performers of Indian classical dances often consider their art as a form of worship. In earlier days their was a practice followed in many part of India by which girls dedicated their complete lives to learn dancing and perform these in temples for pleasing the deity of the temple. As a matter of fact many of the Indian classical dance forms have developed and flourished as a result of such traditions of learning, teaching and performing dances in temples.
Then common people also perform dances on religious festivals to please the god. The festivals on which such dances are performed and the kind of dances vary from one region to another. One most popular of such festival dance is called Garba, dedicated to goddess, which is performed during the nine day long festival of Navratri.
In addition to dancing on special festivals, devotees of different gods sing devotional songs and dance as a part of their daily worship. Two great Hindu Saints, Mirabai and Chaitanya Mahaprabhu, used dancing profusely as a part of their worship. The dancing of this kind by Chaitanya Mahaprabhu, has given rise to a very popular tradition of music and dance called Baul.
Many Hindu gods also practice dancing, perhaps most accomplished dancer among Gods is Lord Shiva, who is also called Nataraja, meaning the "Lord of Dancing". However, the most popular incident of dancing by Gods is the dance called Rasleela, featuring Lord Krishna, when he was a boy of less than 12 years, and a group of ladies from his village.