The games themselves can come in a wide variety of formats, although there's two main variants nowadays: first-person shooters, 'MOBAs'. First-person shooter is self-explanatory, while MOBA stands for Multiplayer Online Battle Arena.
That's the first bit of jargon you'll need to know for esports. Generally, MOBAs are all about two teams playing as heroes with unique abilities, strengths, and weaknesses. The goal is to kill the opponents to gain an advantage and destroy their base on the opposite site of the map. By killing oppponents, they gain gold and experience, which can be used to improve the character.
League of Legends and Dota 2 are the two biggest MOBAs, and have a recognisable top-down perspective. First-person shooters, on the other hand, typically swap between showing a player's perspective and a flying camera when spectating.
In a first-person shooter (often referred to as an FPS), the goal can differ. In some games, like Counter-Strike: Global Offensive, the goal is simply to kill the opposing team. In others, like Overwatch, it can be a 'king of the hill' mode, where teams fight over a location, the first team to control it for a set amount of time wins, or a 'payload' mode, where teams take turns attacking and defending, fighting to push the payload to a final objective.
Other esports can be card games, sport games (the FIFA series is growing in popularity as an esport, with major football clubs having their own FIFA players), strategy games, and fighting games. Sports games are generally the most simple, given they're based on real-world sports, while fighting games are also accessible, given the more obvious back and forth of punches and kicks.
Strategy games, like StarCraft 2, are all about players building armies and trying to defeat each other with optimal unit movement, army composition, and tactical choices. Battles are won and lost of tactical choices and quick movements. Card games, like Hearthstone, challenge players to best one another with monsters and spells, the first person to lose all of their health loses.