Overseas Citizen of India (OCI) Card: Who is eligible to apply and who is not?

The Overseas Citizen of India (OCI) scheme was introduced through an amendment to the Citizenship Act, 1955, in 2005. The scheme provides for the registration of Persons of Indian Origin as Overseas Citizens of India, provided they were citizens of India on or after 26th January 1950, or were eligible to become citizens on that date. What is OCI?