A turn coordinator is an analog flight instrument used in aviation. It is part of the 'six-pack' of standard analog instruments generally provided in analog panel aircraft. It is a gyroscopic instrument, and generally relies on DC electric power to spin its gyroscopes - this way it remains available if the vacuum system fails and the attitude indicator is unavailable.
There are two components to the turn coordinator. The first is either a horizontal line or, in most cases, a representation of an airplane viewed from behind, with its wings as the horizontal line. This indicator will show the degree of roll and yaw rates of change that the airplane is experiencing. Note that it will not show attitude, just the rate at which that attitude is changing. Hash marks on the edges indicate a 'standard rate turn' - when the indicator is aligned with the left or right hash mark, the aircraft will make a complete 360-degree turn in a standard amount of time, usually 2 minutes. Some high-speed aircraft have a 4-minute turn indicator (Concorde was a famous example).
In addition to the turn indicator, there is a slip indicator. This generally consists of a ball in a curved glass tube filled with a damping fluid. When the ball is centered (there are usually lines indicating when this is so) the aircraft is said to be in 'coordinated flight' which means that there are no lateral G forces being exerted on the occupants. Either the airplane is moving in a straight line with its wings level, or it is in a coordinated turn such that the degree of lateral G and bank angle are such that the 'local vertical' remains aligned with 'down' inside the airplane. If the airplane is not coordinated, it will be 'skidding' - lateral forces will be acting on it, be they from the wing loading, surrounding air gusts, or (generally) because the airplane is misaligned. It is turning, but the nose isn't pointed in the direction of its flight relative to the surrounding air.
This instrument is used to ensure that the airplane makes predictable maneuvers, and to make sure that pilot is flying in an efficient, safe and comfortable fashion. If the airplane becomes uncoordinated, the corrective action is to 'step on the ball' - if the ball moves to the left, depressing the left rudder pedal will generally move the airplane's nose back towards coordinated flight, and vice versa.