A transponder is a device which replies to an interrogative transmission (Transmitter-Responder). It is used in civil and military aviation to allow ATC radar signals to trigger an informative broadcast from aircraft in their search area, or as part of IFF systems. They are used to track animals released into the wild. Some are used as navigation beacons.
The term is also used to refer to a paired transmitter and receiver on a satellite, usually because the satellite need only receive a simple confirmation signal that the listener is properly aligned before the associated relay transmitter begins to send data.
Aviation transponders come in several types. In general aviation (light aircraft) in the U.S., the most common is a Mode C or Mode S capable version. These will have three or four controls, at a minimum. The first is a power switch that usually has 'OFF,' 'STANDBY', 'ON' and 'ALT' positions. The first is obvious; the second powers up the circuits but does not enable broadcast; the third sets the transponder to reply with a four-digit octal code, and the last sets it to respond with the code plus the current altitude of the airplane, the 'Mode C' or 'Mode S' of their type names.
The second control is a four-digit code selector. Be careful - when changing codes, avoid 7500, 7600 and 7700 (like, don't even sweep past those settings). They're international distress calls - 7500 means 'hijacking', 7600 means 'radio out' and 7700 means 'general emergency.'
Third, there will be a button labeled 'IDENT' or 'ID'. When pressed, this causes the transponder to send a particular signal that causes your blip on a controller's radar display to 'blossom out.' This way, the controller can ask you to 'squawk IDENT' and see which plane's symbol flares up so he (or she) is sure they know who you are.
Finally, there may be a 'TEST' button which causes the signal lamp to light, indicating that the transponder is (as far as its own little idiot circuits think) working.