Primarily there are 9 types of E-games. There are also many sub-categories. But we would like to make it as simple as possible.
Action Games: the player is in control of and at the center of the action. Sub-categories include "Platformer" (Super Mario Bros.), "Shooter", "Fighting", "Survival" (Fortnite), etc.
Adventure Games: players usually interact with their environment and other characters to solve puzzles with clues to progress the story or gameplay.
Action-Adventure Games: in addition to action part, there is also adventure part (long quests or obstacles that must be conquered using a tool or item collected).
Role-Playing Games (RPGs): gamers are often given choices in this genre that influence the final outcome of the game, which means many RPGs have alternate endings. Sub-categories include "Action RPG" (Diablo), "MMORPG" (World of Warcraft), "Tactical RPG" (Final Fantasy), etc.
Simulation Games: they're all designed to emulate real or fictional reality, to simulate a real situation or event.
Strategy Games: require players to use carefully developed strategy and tactics to overcome challenges. More recently, these type of games have moved from turn-based systems to real-time gameplay. Sub-categories include "Real-time strategy (RTS)" (The Age of Empires), "Tower defense" (Plants vs Zombies), etc.
Sports Games: simulate sports like golf, football, basketball, baseball, and soccer. They can also include Olympic sports like skiing, and even pub sports like darts and pool. Racing is also a popular sub-category.
Puzzle Games: puzzle or logic games usually take place on a single screen or playfield and require the player to solve a problem to advance the action.
Idle Games: usually small games and do not take long time to play. Sub-categories include: "Party game" (Mario Party), "Board game" (Chess), "Card game" (Pokemon card game), "Educational game", "Exergame" (exercise game).