Which Luna Display port type do I need to purchase?

It depends on the model and year of your Mac. Many older Mac models have MiniDisplay / Thunderbolt 2 ports, and most of the latest Mac models have only Thunderbolt 3 / USB-C ports. 

Check out Luna Display guide for more information, here -- https://lunadisplay.com/pages/choosing-your-luna

As well as general port information for different Mac models through Apple Support pages - https://support.apple.com/en-us/HT204154