Plain English Meaning
A spinning wheel inside the instrument that is mounted sideways instead of upright. Because it is mounted that way, it reacts to the airplane turning left or right and drives the turn needle.
Full Definition
A gyroscope whose spin axis is mounted in the horizontal plane, so its rigidity in space is referenced sideways rather than up and down. In a turn-and-slip indicator, the horizontal gyro spins on a lateral axis and is gimbaled so that yawing of the aircraft (movement about the vertical axis) tilts the gyro through precession, which moves the needle to indicate rate of turn.