Generally it works like this:
You apply voltage to a component .. and based on it's inner resistance, some current will go through it.
Many components do not react in linear way to voltage, diodes are one of them. Anyway for working ranges of laser diodes, the reaction is almost linear. You increase the voltage, the current will increase in linear fashion .. and so will the power.
There is usually some minimum current - called lasing treshold. There is usually some maximum voltage, and maximum current, neither of which should be exceeded. There is usually also some maximum temperature, which should not be exceeded.
Then there is some 'sweet spot' .. current at which the diode works best.
Now why you need constant current driver and how it works ? Diode changes it's 'inner resistance' based on temperature. As it gets hot, the inner resistance goes down. If you applied constant voltage to it (which for example common AC/DC converter do), the more and more current would go through the diode .. increasing temperature, decreasing resistance and increasing current again .. up to the point the diode is killed. This is called thermal runaway.
Also diode output depends on current, not voltage. So to get safe and constant output, you need constant current driver. Such driver has feedback loop inside it, which will lower the output voltage, if output current is higher then preset current .. and increase the voltage, if it's lower.
But unless the diode is near it's runaway state, any preset current will be associated with specific voltage. For example 445nm diodes, if you want about 1W of power, you need about 1A of current, which is usually about 4.5V of voltage. So it's all interconnected.
The driver has to be constant current. Has to be able to generate specific current .. but in specific voltage range. You can have constant current driver, which can generate 1A, but it's output voltage could go only up to it's source voltage. And you want to use Lion battery with 3.7V. Such driver wont be able to generate 4.5V needed to get 1A from 445nm diode, thus wont be able to feed the diode. But it could be used on red diode for example, as their working voltage is lower at the same current.
You need driver which meets both current and voltage requirements, with given source. Also the power should be considered, but that depends on voltage and current, so if some driver is designed to given current and voltage, it's also designed to respective power. This could imply colling policy might be necessary though. That should be part of driver design .. like for this and this power you have to use heat-sink this and this big.
As you have to test both voltage and current .. for testing and setting the driver you will need what's called test-load. It's simple circuit which simulates the specific diode reaction to voltage. It consists of diodes and resistors, but it's different for each diode type. Setting the driver with resistor as test load, or different diode (like your LEDs) wont work. It could work if the LED was same color as intended laser diode, and could take same current, as laser diodes are basically LED with specific construction. But it's unlikely that such LED would exist if you are building high power laser.
