Leagues and tournaments would be set up similarly to your physical sports leagues.
Running leagues vs tournaments is solely up to how much you're ready to take on. Some departments offer all the programming in house, meaning they have a space for players to meet up and equipment for them to use (or for players to bring their own and use). Some offer leagues where the players play off site and report scores to the person in charge. Either way, organization can be as simple as a spreadsheet to keep track of games or the utilization of a company or app to help facilitate. Take into consideration that you can do a tournament in a smaller timespan like over a weekend where your league can occur over months.