Each dungeon can be completed in Normal or Veteran mode.
Veteran mode is unlocked once you reach level 50 and increases the difficulty for the group by making the mobs and bosses even tougher. In turn, veteran mode dungeon runs reward players with better quality gear and unique achievements and collectibles.
To take on a dungeon in veteran mode, select the veteran option in the Group & Activity Finder menu before entering the dungeon.
You can also specify normal or veteran versions of a dungeon when queuing.
Thanks to ESO Wiki for info in detail.

Finally, when taking on a veteran dungeon, you can attempt to beat the final boss on Hard Mode.
This will offer even greater rewards (and a greater challenge)! To find out the conditions for hard mode within a unique dungeon, check the Dungeons section of your Achievements menu (found within your Journal). To activate hard mode, some bosses only require you to activate a nearby scroll, while others require you to defeat the boss under unique conditions.
(Note DLC required from crown store for DLC content)

Every zone has a large public dungeon. Whilst many of these can be cleared on your own with the correct build and gear, they are much more difficult to take on solo than the smaller public dungeons. What sets the large public dungeons apart from the small ones — apart from the fact that these dungeons have much more enemies in them — are group challenges, and a greater number of dungeon champions, who are named dungeon bosses and a part of public dungeon achievements.

Aldmeri Dominion Public Dungeons

  • Toothmaul Gully – Located in eastern Auridon.
  • Root Sunder Ruins – Located between Elden Root and Haven, in Grahtwood.
  • Rulanyil’s Fall – Located in central Greenshade, west and slightly north of Marbruk, on the other side of the mountains.
  • Crimson Cove – Located north and east of Velyn Harbor, on the coast in Malabal Tor.
  • The Vile Manse – Located in the northwestern corner of Reaper’s March.

 

Daggerfall Covenant Public Dungeons

  • Bad Man’s Hallows – Located just west of Daggerfall City, on the coast in Glenumbra.
  • Bonesnap Ruins – Located in western central Stormhaven, west and slightly north of Wayrest.
  • Obsidian Scar – Located in central Rivenspire, northeast of Shornhelm.
  • Lost City of the Na-Totambu – Located in central eastern Alik’r Desert, directly west of Satakalaam.
  • Razak’s Wheel – Located in the very southwestern corner of Bangkorai.

 

Ebonheart Pact Public Dungeons

 

Coldharbour Public Dungeon

Wrothgar Public Dungeons