How to set a positive and negative current from arduino?
Use something known as a H bridge.  That's basically two positive only outputs.  You connect the device between them.  Let's say each output can drive between 0 and 10 V.  The voltage between the two outputs can then vary &plusmn;10 V.

This only works if both sides of the device to be powered can float relative to ground.  However, that is probably the case for your Peltier.  Check the datasheet of the laser module to make sure both lines can float arbitrarily relative to the other connections.

A Peltier is one device that should NOT be run with pulses.  However, pulses are the easiest and most efficient way for the power drive system to work.  The solution is to add some filtering between the raw H bridge outputs and the Peltier.  These would usually take the form of a series inductor followed by a capacitor to ground on each of the two outputs.  Another way to look at this is that you're making a switching power supply to run the Peltier from.

I go into more details on driving Peltiers at https://electronics.stackexchange.com/a/28637/4512 and https://electronics.stackexchange.com/a/118543/4512.