There are two types of turn indicators available for aircraft. These are: the Turn and Bank Indicator (T/B) and the Turn Coordinator (T/C). Both are driven by a gyro and indicate the rate of turn but the turn coordinator can also indicate rate of roll. The difference between these are where the T/B has a needle indicator and the T/C has an aircraft picture on the face of the instrument.
Neither of these instrument give an direct indication of bank angle. For that you will need the speed of the aircraft and rate of turn to calculate the bank angle, or have an attitude indicator installed.
Also note that these indicators are part of the basic six instruments in an aircraft the pilot needs for legal instrument flying. Notice that nowadays you see more and more EFIS panels where the original T/C and T/B are used as standby and backup in case of failure of the main EFIS.
Turn indicators use a rate gyro (see image) to detect how fast the aircraft is changing direction. The T/B has a vertical needle and a slip ball; note that the Turn Coodinator has an aircraft picture. Both indicate a two minute turn, thus a 360° turn is completed in two minutes or put differently: three degrees/second (3°/s, not that famous group) which is a rate one turn.
The coordination ball indicates the direction of the g-forces and its the pilots job to keep the ball centered by 'stepping on the ball' with the rudder pedals. Thus: ball to the right -> apply right rudder or trim (and less left rudder) and vice versa.
Rate gyro operation
This gyro has only one plane of freedom where it can move (not the plane of rotation). The tilting of the vertical axis though the aircraft causes precession and this is used to indicate the rate of turn about that axis. The amount of tilt (linear with the rate of turn) is balanced against a restraining spring.
T/B or T/C
In a T/B the gyro is mounted in such a way that the rotation (or turn) axis is from wingtip to wingtip. In the T/C the turn axis gimbal is mounted on a 30° angle upward from the longitudinal axis of the aircraft, and this gives it the ability to sense rate of roll.
The springs used in the rate gyro give proper indication of rate of turn by being able to adjust the tension of the spring against the precession force of the gyro, without the springs the gyro would just bounce around. This is then indicated on the face place.
Due to air turbulence and aircraft movement by the pilot there will always be a force on the gyro causing precession and thus needle movement. To stabilize this indication the gyro has internal damping devices limiting these movements and presenting a stable indication to the pilot.
The gimbal ring reaches a mechanical stop when the aircraft turns about 20°/second. And with only one gimbal ring the gyro will not topple as with the attitude and heading gyro's.
These are the result of mechanical design and limits due to contruction. Indication due to turbulence is not really an error by instrument design but this will cause erratic movement only.
When the gyro rotor speed is not within limits, the indication will suffer and be much slower. When the rotor speed is too high it will result in an excessive rate of turn as the precession forces will be greater than the preset tension of the calibrated springs. The instrument will look very much alive in this situation
Another error occurs when the aircraft subjected to a yaw movement together with positive g-forces, the indicator will then overread. It is most noticeable when pulling out of a dive with the aircraft out of balance or with an unbalanced steep turn.