The Hindi word for "princess" is "raajkumaari," which is written "राजकुमारी." Many different titles were used for royalty during the history of India, depending on the culture, religion and language of the area. Female Indian rulers were often titled "Maharani" or "Begum."
"Maharani" was used across India, especially for Hindu rulers, while "Begum" was used during the Mughal Empire. There is no such thing as a Native American princess, because the tribes did not have royalty. It is popular lore for some that a person's great-grandmother or another ancestor was a Cherokee Indian princess, but it is not true.