You need to send in enough current into the Base to allow the collector-emitter path to go into saturation; that is, to be turned full on.
One way to do that is put as much current into the base as the arduino can live with.
The voltage across the base-emitter junction is usually about 1 diode drop, or 0.7V.
The arduino can put out 20mA pretty safely. A resistor in series from arduino to base limits the current.
With arduino pin at 5v and base at 0.7V, there is 4.3V across the resistor.
Using ohm's law, V=IR, or V/I = R, we can determine the resistor value:
(5V-0.7V)/20mA = 215 ohm
If your transistor has a gain (Hfe) of 50, then 20mA into the base will allow 1000mA thru the collector/emitter.
If the device you are driving has some resistance/impedance, like a motor, then that becomes the current limiting factor, and not the transistor, which is the situation you want when using a transistor as a switch.