Does a specific laser module have a specific amount of power needed?
The short answer is yes, different LASER diodes will have different operating currents.

A diode is current driven device, which without going into too much detail, roughly means you can't just apply a voltage and expect it to work properly. In particular LASER diodes are very sensitive to overcurrent and get damaged easily, so more care must be taken with them (e.g. compared with an LED) You must have current controlling element in series with it.
Depending on what this element is, it may be safe to apply a higher voltage to it, or it may not be.  

For example if the current control element is just a resistor, then applying a higher voltage will probably break the diode, whereas if it's a constant current driver IC, as long as it's within it's operating range it will make no difference (the current will remain constant, hence the name)  

Resistors are a bad way to do things, but are commonly used in the cheap Chinese LASER pointers. Better designs use a control IC which provides a constant current. The best also use optical feedback as well to control power to the diode. Here are some example LASER driver ICs.

You would need to examine your LASER pointers circuit and determine how it is being driven, find the values/datasheets of any components present and/or measure the currents/voltages with a multimeter.   

A typical diode looks like this - note the 3 pins, the third is the photodiode used for optical feedback:



Here are a couple of example datasheets, take note of the difference in driving currents:

Low Power
Higher Power  

Also Sam's LASER FAQ has tons of info on LASER diodes (plus other LASERs) and how to use them.  

Here are a couple of example circuits:

Basic constant current circuit:



Constant current and opto feedback diagram:



Opto feedback circuit example: