In addition to the “public” dungeons, there are also sixteen group dungeons throughout Tamriel. There's an NPC guild called the Undaunted that are based primarily on exploring these dungeons, and joining their ranks gives the player access to a skill tree that contains abilities that are essential for completing the dungeons. You'll want a full party of four players if you want to tackle dungeon content, and though there's some built-in group finding capability, it's a lot easier to just organize a party on your own.
Since I was in the Ebonheart Pact, the first dungeon I was able to tackle was the Fungal Grotto in Stonefalls. Like other MMOs, the dungeons in ESO contain a number of sets of trash mobs to clear out and several boss encounters. Unlike other MMOs, however, the bosses are immediately complex, requiring careful interruption of abilities and multitasking, especially when the bosses summon friends. In fact, just about every boss in the place either started with a pack of allies or was able to summon them. Even the giant mudcrab boss. Yes, there's a giant mudcrab boss.
The final boss in the grotto was a giant Dreugh who is able to vortex players into melee range and can immediately wipe your party with a large AOE attack if you're not careful. Being able to escape the range of his attacks is key. He also periodically summons an army of mudcrabs that need to be killed quickly. Completing the dungeon's quest grants you a skill point and unlocks two dungeons that exist in the enemy alliances for exploration. Since you can't explore enemy lands until you've completed the storyline for your faction, this is a welcome feature.
After my time with the beta, I wasn't too impressed, but the retail version of Elder Scrolls Online was able to win me over in the end. It still starts out a little too slowly for my tastes (I probably put 20 hours in before I did my first dungeon), but it's well worth a second look.