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 centshttp://rcarduino.blogspot.com/2012/10/arduino-serial-servos-20-servos-4-pins.html
18 servos using a pin per servo (20 if you had enough pins)http://rcarduino.blogspot.com/2012/11/how-to-read-rc-channels-rcarduinofastlib.html
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.