generator is your bottle with your yeast mixture in a DIY setup.
reactor is a thing people build that mixes the CO2 into the water, which typically involves a diffuser of some kind, which is an airstone, powerhead, anything that breaks the large bubbles of CO2 into small bubbles.
I've seen reactors built that are basically a tube with tank water pumped through it and stuff like bio balls and whatnot inside with a CO2 hose hooked up and as the water gets pumped through the tube it picks up the big CO2 bubbles, then they break into smaller bubbles as they run over the bio balls and whatnot. IME they are mainly for larger tanks with more complicated filtration systems.