I'm just trying to speak plainly here so don't take offense if it sounds over-simplified. I'm trying trying to be explicit.
The PCB is the game board. It's the brain of the game. There is usually one main connector on one end of the board that most (or all) of the wires connect into. This is called an "edge connector" because it's on the edge and it's a connector.
The massive plug that goes into it is called the "wiring harness" because it's a harness that holds the vast majority of wires going into the board. These wires contain power, audio output, video output, and leads for all the switches in the control panel/coin door/etc. Sometimes a game has too many controls or special power requirements so not all the wires will feed into the harness. For this there should be an extra plug or two on top of the board.
There is a basic standard for the majority of games made in probably the last 20 years called the JAMMA (Japanese Arcade Machine Manufacturers Association IIRC) standard that calls for a unfiorm edge connector and harness. That way games can be interchanged by just pulling one out and putting a different one in. Anything older than that will have a custom edge connector and harness.
There shouldn't be any soldering needed if you have a compatible board and harness. Everything just plus in. Otherwise, adaptors can be built or bought if you need to convert from one harness to another. There should never be a need to solder onto the board, itself.
How's that for a start?