Official Aviation Definition
An aircraft transponder is an airborne transmitter-receiver that automatically replies to ground secondary surveillance radar (SSR) interrogations. Mode A transmits identity (squawk code), Mode C transmits pressure altitude, and Mode S transmits rich data including aircraft registration and flight ID.
In Simple Words
A radio in the aircraft that talks to ground radar towers, telling controllers who you are and what altitude you are flying at.
Real-World Flight Example
The crew switched the transponder from Standby to Altitude Reporting Mode (Mode C) as they crossed the runway hold short line.