Fan turns on without key being in the ignition. What does that mean/how to fix?

The main cooling fan for the engine is a fan hub withe silicon gel in the center of the hub, the fan that you hear staying on is probably the condenser fan. Most likely the fan relay is stuck, it will need to be properly diagnosed before just jumping to conclusions. It's not a complex problem a shop familiar with Buick or GM products would probably charge an hour to an hour and a half diagnostic time to diagnose the problem the time would go toward fixing the problem and would most likely be fixed well quicker than that.