While Bajdi is correct, a Arduino UNO can also control 18 Servos using my servo library or 20 Servos using only 4 pins if using my Serial Servo library.
20 Servos 4 pins - total cost - about 60 cents
18 servos using a pin per servo (20 if you had enough pins)
The challenge you will have whatever you decide to do will be supplying power to all of the servos.
Generating the control signals for 20 servos is a piece of cake for an Arduino UNO, it uses less than one percent of the Arduino processing power.
Powering a single servo requires more power than an Arduino can deliver, figure out your power and then worry about the board to use.
Duane B