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.