Group Dungeons are instanced zones intended to be completed by coordinated groups of four players. They can be accessed by any faction and at any level. They feature detailed quests and grant a Skill Point and Achievements upon completion. Each Group Dungeon has a Normal and a Veteran mode.
Normal dungeons used to follow the character level progression through the quest zones, but since the introduction of One Tamriel, there is now a recommended character level based on the difficulty of the dungeon. You can enter the dungeons from the world map at any level, however you will not be able to enter via Dungeon Finder if you are below the recommended level for the dungeon.
Veteran dungeons are available at character level 50. They feature more difficult mechanics and drop more powerful loot, including Monster Helms from the final boss of each dungeon. See the Veteran Content page for more information. Some are locked behind higher CP levels when using Dungeon Finder, but unlike with Normal dungeons, you cannot enter them via the world map either.
(Edit: You can enter Veteran Dungeons from the world map, the same way you would enter a Normal dungeon. You must first make sure you set your group difficulty to Veteran. This must be done prior to any group member entering the dungeon or it will not allow you to switch,)
New dungeons are usually added when new DLC is released. These are intended to be challenging to max level players, and generally should not be attempted by new players.
Base Game Group Dungeons
Dragon Bones DLC Group Dungeons
Horns of the Reach DLC Group Dungeons
Imperial City DLC Group Dungeons
Shadows of the Hist DLC Group Dungeons
Wolfhunter DLC Group Dungeons
Wrathstone DLC Group Dungeons