A form of United States currency, specifically a Federal Reserve Note, which is the paper currency presently used by the United States. The ink on the reverse side of Federal Reserve Notes being green is what led to the nick name "greenback". To a lesser extent, Silver Certificates and United States Notes also had green backs, but the seal and serial numbers on the front were printed in blue instead of green ink. They were last issued in 1957 and 1966, respectively. Your parents probably remember them.
'Greenbacks' have been printed since before the turn of the 20th century, when the size of the paper currency was printed on was about 1/3 wider and 1/3 longer than its present size.