If you do not like watching games that feature first-person shooters, then there are fighting game tournaments, racing game tournaments, MOBAs and much more. Just like there are multiple sports options for the discerning sports bettor, there are multiple video games that people compete in. League of Legends, for example, has leagues with franchised teams, much like the NBA.
Most of the time, they take a tournament or league format and run with it. Esports competitions can range between tournaments holding 10 people or thousands of competitors in a sports arena. To put it in the absolute simplest terms, esports essentially refers to video game tournaments, leagues, and competitions.