The Barred Parakeet, also called Catherine's Parakeet and, particularly in aviculture, the Lineolated Parakeet (Linnies), comes from the Central America and northern South America.
The natural colour is olive green with dark edges to feathers and dark barring or spots on the sides; many other colour mutations have been bred in aviculture.
The Grey-cheeked Parakeet is also sometimes called the Fire-winged, as in its scientific name. In addition to its grey cheeks it has a greenish-blue cap and a bright orange bar under its wing (just visible as a small orange spot in two of the pictures below) which gives rise to its scientific and less common names. It comes from west Ecuador and Peru.
The Plain Parakeet has several shades of green with a reddish bill. Its wings are darker green than its head and body and there is sometimes a blue on its wingtips and tail.
In fact, it is not as plain as its name would indicate. It is native to Brazil.