Is there a device that I can put in between a motor and an AA battery that will only allow current to pass from the battery to the motor when the Arduino gives a signal? (I have attached a drawing).
I need the arduino to drive a small pager motor in one direction.
Sure there is. The ? three terminal box you drew can be a transistor that will switch the motor on or off. You can choice to use a simple digital output command to turn it on or off, or use the analogwrite command that would allow variable speed output, from off to full speed. The motor should be mounted on the other side between the battery + and the transistor collector lead. Also the battery - terminal must also wire to the Arduino ground pin.
Here is a drawing from the Arduino Playground that controls a DC solenoid, but just think of the solenoid connections as your motor connections. All the parts shown have a purpose so you would be best just to copy it completely:
If you are looking for a motor driver then heres a good one: http://www.solarbotics.com/products/k_cmd/
it can drive 2 motors at 6 to 50V, at up to 4A total output current. It is a nice little kit that delivers nice current for its price. It also has an onboard voltage regulator so its the perfect thing to connect between you batteries, arduino, and motors ;D
For a small pager motor in one direction, i would think it can be driven directly from the arduino output pin.
If it uses 1.5V, a single AA battery, then a series resistor that limits the current is probably the only extra component needed. This should be designed to give a 3-3.5V drop at the specified operating current of the motor.