Dungeons allow players to group up for challenging instanced content. There are three different tiers of dungeons: Normal, Challenging and Heroic. All dungeons are available for parties of up to 6 members, and you can enter them through their respective portals in the game world or, most commonly used, through Dungeon Lobby (F8) which allows you to recruit manually on chat or LFP (automatching).
There are 2 modes which you can choose when looking for a dungeon: Easy Mode and Hard Mode. Hard Mode offers higher amounts of certain common rewards than Easy Mode as well as exclusive rewards such as outfits, psyches and other valuable items, however it requires much better gear and experience with the dungeon, since failing the mechanics will result in the boss enraging, defeating the party. On the other hand, failing the mechanics in Easy Mode will still cause an enrage, but it won't instantly defeat the party. The bosses also have their stats significantly reduced in Easy Mode.
In addition, there are other instanced zones, usually called "raids", which you can't enter through the Dungeon Lobby and are made for a 12 people alliance. You can identify them by their Legendary tier.