Servo systems do the same thing in every videotape machine, but differ so much in design that it is hard to give more detail that what has been presented already. I will add some material that will help you understand servos a bit more.
The analog servo circuits in older machines make heavy use of a circuit called a Sample and Hold Phase detector. This circuit requires two inputs: an input pulse and a sample pulse. The input pulse starts a linear ramp generator that produces a smooth ramp with a period a bit less than the sample pulse repetition rate. The sample pulse opens a switch that charges a capacitor to the instantaneous voltage of the ramp at the moment of the pulse. The capacitor stores this voltage, which is buffered and presented to the outside world as an error signal. If the sample pulse occurs a little earlier than normal, the sampled voltage is a bit lower (Assuming a positive-going ramp.). If the sample pulse occurs a bit late, the sampled voltage is higher than normal. The error voltage is used to change a motor speed, or retune a VCO, etc.
Another circuit sometimes used for measuring error in a servo circuit was a frequency discriminator The discriminator used a special kind of tuned cirucit to measure the frequency of a signal. The output of the tuned circuit fed a pair of rectifiers. If the frequency of the signal was correct, the tuned circuit's output would be balanced, and the signals at the outputs of the rectifiers would cancel. If the frequency was high or low, the balance of the tuned circuit would be upset. The output of the two rectifiers would not be equal, and their sum would be an error voltage proportionate to the frequency error. Although not usually used directly for electromechanical control, the frequency discriminator was used to help produce reference signals from sync, etc.
Newer machines measure error by counting digitally generated pulses between the input pulse and the sample pulse. A change in the count causes the servo microcontroller (Or a special digital servo chip or chips in earlier machines.) to issue a error correction adjustment to a motor speed, etc.
drum and capstan servos
The drum and capstan servos are actually two separate loops: the speed servo and the phase servo. The speed servo is usually a sample and hold type circuit that compares a motor tachometer pulse to itself through a fixed delay. When the speed is correct, the delay puts the sample pulse in the center of the ramp. This circuit gets the motor up to speed, and keeps the speed approximately correct. The phase servo compares a reference signal to a variable signal generated by motion of the controlled device, and issues a speed change to bring the controlled device into an exact state of phase lock. You could say that the phase servo knows the exact shaft angle of the motor at any given moment.
All servo systems need a reference to carry out their correction activities. During record, the drum servo uses vertical sync as a reference, and the capstan servo uses the drum tachometer pulse as a reference. This ensures these motors run at constant speed, and are locked to each other. On playback, the drum servo uses reference vertical as it's reference, and compares it to off-tape vertical. The capstan servo uses the drum tachometer as a reference, and brings the control track into phase with it. Much of this phase comparison is done using sample-and-hold circuits, or pulse counting discriminators. Some early servos, such as the Ampex Intersync also used FM discriminators to measure the frequency error of a tachometer signal.
Most modern VTR's employ some form of electronically commutated DC motor for those motors under servo control, especially the drum and capstan motors. Older machines used a variety of interesting motor control techniques, as the technology for electronically commutated motors was not yet available. Early quad machines often used polyphase AC motors, of two or three phases. Digital dividers and power amplifiers provided the sometimes substantial AC power needed to operate these motors. This made the best use of existing motor technology as no commutators, slip rings or active electronics within the motor were required.
driving the video head drum via a belt
Another popular speed control method employed a constant-speed motor driving the video head drum via a belt. The pulley diameters were chosen such that the head would rotate slightly faster than required. The belt was of special design, and could slip slightly without damaging itself or the pulleys. An iron armature was mounted on the head drum shaft, and was surrounded by a DC electromagnet.
When current flowed through the electromagnet, the magnetic field would provide a braking force to the armature, and slow down the head drum. By varying the current, the necessary speed control could be effected. Servo failures in these machines were often easy to recognize, especially if the servo called for excessive slowing. There would be an unearthly screeching noise from the machine, followed by the smell of burning rubber! This scheme was known as the Drum Brake.
The last thing to mention in the servo section is the control track record/play circuit. For quadruplex, the control track is a 240 Hz sine wave on record. Since it is a direct recording, the playback signal is a distorted, double-humped signal that actually represents the rate of change of the original sine wave. A filter circuit in the servo section of the machine converts it into a clean sine wave, or a pulse.
All other VTR formats I know of record a simple square wave. It is reproduced as positive and negative going spikes corresponding to the leading and trailing edges of the original square wave. Also, most professional formats record an extra pulse to denote the beginning of a new frame.