To connect a simple toy circuit and trigger with an arduino?
You'll probably want to use a transistor to switch it on/off. You could just use the Arduino's digital pin but they can only supply a limited amount of current. Transistors can work with larger currents. The Arduino switches the transistor and it powers the external circuit.

There are a couple of existing questions that may help you.


How to switch an external circuit with Arduino?
How can higher current devices (motors, solenoids, lights, etc.) be controlled by an Arduino?


jippie's answer in the first link shows an example schematic for hooking up a transistor with a bunch of LEDs but you would replace them with your own circuit: