The history and origins of Bharatanatyam are very complicated. There are different speculations about how the dance form came to be and what style is the original. It is very strongly tied to the Hindu religion, and is very deeply rooted in South Indian culture. The religious aspect claims the dance was created by Lord Shiva, one of the most powerful Gods in Hinduism. Otherwise it can be traced to different teachers. Initially it started as a very religious dance. Devadasis, or servants of God, were the ones who were said to dance in temples for religious occassions and as their way of worshipping. During the medieval era, it became a court dancer's dance, an erotically evocative style. So gradually different styles started emerging, and people of different religions started learning. It is now practiced as a cultural dance and can be learned by anyone.
It is very similar to ballet because it is expressive a dance drama and very expressive. The main difference however, as you may have noticed in the videos, is that it is also very structured. Balled is a very fluid dance style, while Bharatanatyam is very structured and beat-oriented. Female dancers are more common, for the simple reason that it is a very expressive dance style and requires a lot of grace. Male dancers do exist, and the dance can be made masculine if the right part is assigned. If a male part exists but no male dancer is available, a female can dress up as a man. And sometimes vice versa.
India has always been known for its rich culture. The different regions of the sub-continent provide stark differences in language, food, clothing, and even dance styles. You are now familiar with Bharatanatyam. It originated in the southern-most state of Tamil Nadu.
A little further up is Kerala, where the traditional styles of Kathakali and Mohiniattam originate.
Next in the west is Orissa, where the dance of Odissi originates.
The next major dances come from the north. The classical style prominent in the north is Kathak. This style has heavy Muslim influence and is done to the Hindustani music style. Folk dances in the north are also very popular. In Gujurat and Rajasthan a mostly desert/village area, the dances are Garba, which is mostly done by girls, and Raas, which is interactive for girls and boys. Raas is said to be the dance that Krishna, a popular Hindu God, created.
And the furthest north in Punjab is a very upbeat dance style called Bhangra. It is very masculine but does have a female style too. Bhangra is like India's hip-hop.