Most of the time just knowing how many the coils the motor has and āhow bigā the motor is, should give you decent indication of what to choose.
Every coil has 2 wires, so if your stepper has 4 wires, it has 2-coils. This is the most common setup btw, and 99% of the boards support it, so you can choose between A4988, TMC2209 and some others.
About the āsizeā of the steppers you donāt really need to mind, lasers have generally little steppers in them. NEMA 17 or so and the drivers mentioned above runs them pretty well.
The difference between the drivers mentioned above? The TMC is a little bit quieter and such, and for advanced users has a serial port to get more than 16 microsteps out of it, but I think that doesnāt matter for the average laser user. If you use marlin, the TMC (in serial mode) also has āsensorlessā homing (must be enabled and firmware must support it) in which it basically rams into the endstop with very little force and āsensesā that, immediately stopping and giving feedback to the controller. Most likely your laser already has endstops, which are much more precise then ārammingā into a wall, so you wonāt even need this. Overall, the TMC has more features, but thatās about it I guess. If the price difference doesnāt hurt you, go for the TMC. Less noise, less vibrations - less vibrations, better lasing 
About the controller, go with the 32Bit boards we mentioned above. They all supoort A4988 and TMC chips, donāt worry about that. In the 3D community, theyāre pretty common. Just look at the socket of the drivers and youāll immediately know.
The firmware you choose after it arrives, you flash it with your own PC and tweak it to your needs. Thereās documentation and tutorials all over the internet. However, if you need help, feel free to PM me!
Hope that helps out clearing some questions 