Trooper is the equivalent rank to private in a regiment with a cavalry tradition in the British Army and many other Commonwealth armies, including those of Australia, Canada, South Africa and New Zealand; it is also used by the Irish Army.
Cavalry units are organised into squadrons, and squadrons are divided into troops. A trooper is a member of a troop. "Trooper" can also be used as a nickname to mean any cavalry soldier (but normally not for an officer).
In the United States Cavalry and airborne forces, "trooper" is a nickname that has traditionally general term for any enlisted soldier, and is not a rank.