Ordinary brush type DC motors come in several flavors. Series wound motors (also called universal wound because they can run on AC) have the field windings wired in series with the armature, so that the torque produced is proportional to the square of the current (the armature current reacts against the equal field current). Since these produce torque in the same direction, regardless of the direction of current (hence the usability on AC). So these are not usable as servo motors.
Shunt wound motors have the field winding either wired in parallel with the armature, or excited by a separate current, entirely. The separately excited shunt wound motors can be used as servo motors, since their torque is essentially proportional to armature current and their speed is approximately proportional to armature voltage.
Permanent magnet field motors are very similar in character to separately excited wound field motors, since their field's magnetic strength is not related to armature current. They are commonly used as servo motors.
Any motor that can produce torque in either direction can, in theory, be made into a servo motor if you can measure its speed to be used by the servo loop controller. Being able to measure the torque is very handy, also, so motors that have torque proportional to armature current make this easy.
Strictly speaking, a servo is a motor application, not a type of motor.