The APs are just connected to a switch each (in my case, I use switches with PoE, so the APs get power from the switch, but that's totally optional).
The switches then backhaul to a central switch, with later connects to a router and then a cable modem. All totally conventional SOHO networking stuff. All of mine happens to Ubiquiti except for the Arris cable modem, but it could be mix-and-match just fine.
The controller software I run (intermittently) on my Mac laptop. It is only needed for "adoption" and configuration of the devices (and if you wanted to have any live logging or live guest wifi access portal [like in a hotel]).
Basic wireless functionality, including WPA2 authentication, is handled on the APs themselves without the need for my Mac (and the controller software) to even be at home or be turned on. It's completely standalone at run time. (Put another way, I put the same system in at my parents 700 miles away and the controller is on my Mac here as well. I visit 2-3x per year; other times, they run standalone, including recovering from power outages, etc.)
In short, it works exactly like you probably hope it does.