Rita Moreno was not born with that name. Her birth name was Rosa Dolores Alverio. She was born in Puerto Rico.
Her parents got divorced and her mother moved to New York. Her mother got a job sewing clothes in a sweatshop. After she got settled and learned some English, she went back to Puerto Rico and got her daughter and brought her to the US. Rosa/Rita was five years old at that time and did not speak any English at all. She entered Kindergarten that year.
Moreno was later seen dancing (at her home) by a friend who connected her to a man who taught her dance. He was the uncle of the actress Rita Hayworth. Through him, she learned dance and then moved on to acting on radio programs. She got her first Broadway role at age 13.
To hear Rita Moreno talk about her childhood and more, please follow the emmytvlegends.org link below.