Dungeons are locations with a high concentration of enemies and often at least one named boss. Defeating the named boss(es) is usually required for the dungeon to be considered "cleared", or to complete an associated quest.
Killing enemies and healing damage in many (but not all) dungeons contributes to progress for a group of Dungeon Achievements.
The following place types are all considered to be types of dungeons:
Delves, dungeons where you may meet other players not in your group
Public Dungeons, more difficult than delves, designed for group play but still public
Group Delves, dungeons where you will only see other players in your group
Group Dungeons, more difficult than Group Delves, where you will only see other players in your group
Trials, elite dungeons designed for large groups
Arenas, dungeons with round-based combat challenges- Most interior zones found at Points of Interest and Unmarked Locations
A full list of Dungeons can be found in the associated category.