For a Normal person - Kathak looks like dance with fast paced movements mostly with pirouettes and footworks. It might look simple, but in reality performing requires decade long practice. Since it has most graceful and simple movement - the style demands extreme complexity in rhythm and speed with the movement. This needs to be attained without losing the beauty.
Original name Kathak is derived from Katha kaar or story tellers. It used to be a good time when these professional story tellers told the stories of Hindu gods in temple. AS the time passed, they added some movements. During 12 - 15 century - in the Muslim Indian rule many chose to entertain the royals and rich people through their artistic works of excellent movement and foot work. Many a time it used to be highly competitive between dancers to please a Sultan or Nawab to win accolades.
The current form of Kathak divides into three families or schools.
Jaipur Gharana from city of Jaipur - where rhythm is given more importance. They have created some amazing rhythmical phrases and rapping sequences. They also tell the story in a simpler way.
Lucknow Gharana - This has gelled out from the patronage of Nawab of Oudh, who was a great admirer of art and encouraged sensualities of abhinaya. This graceful mentorship continued to make the Lucknow style very graceful.
Benaras Gharana - This is the powerful, strong dance presentation style. They elaborate the story lines in an excellent way.