What is "Derby"?
The Derby is a horse race. Or to be precise, it is the title of the richest and the most prestigious race in a country.

Why is it called the Indian Derby?
Every nation's best horse race is called the "Derby". In England, it's named after the racetrack where it is run (Epsom Derby), and in the U.S. it carries the name of the state in which it is run (Kentucky Derby). In Japan and India, it's called the Japan Derby and the Indian Derby respectively.