NRI means a Non Resident Indian.
Many times, people leave their home country and travel to reside in another country. This could be due to work or family. These people are still citizens of their original country, but they do not reside there.
Similarly, Indians who do not reside in India are called NRI, which stands for Non Resident Indians. A person is considered an Indian citizen, if he has an Indian passport and resides in India permanently. If an Indian citizen lives outside of India for more than 6 months out of the year, then he is considered an NRI.
However, if an NRI gives up his Indian citizenship, and accepts the citizenship of the country he resides in, then he will no longer be considered an Indian citizen, or an NRI. He would then just become a PIO, i.e. person of Indian origin.