In the context of gaming, clans are used as a means for players across the world to team up together. A clan is simply a team but often has a more military-style setting to it. Clans are most often used in competitive multiplayer games where players will work together.
Many online games have built-in clan support which will allow players to display a clan tag in front of their username so that other players can identify the group they belong to. For team deathmatch games, you will often enter a lobby where you are up against a team of people with the same clan tag.
In standard deathmatch games, people often use clan tags as a way to show quality and skill. Some clans will only allow skilled players into their teams. If you are determined to excel in a game, it is best to figure out the best clan names there are and work to become part of this team.