Dungeons in Final Fantasy XIV: A Realm Reborn are a bit different than other MMOs in a lot of ways, but similar as well. First and foremost they're rather easy, designed for four players (one tank, one healer, and two DPS) and are very lenient. That isn't to say it's a cake walk. If you pull too many enemies or don't tank the enemies properly then your group will wipe.
One unique thing to know about dungeons is that you cannot receive a /tell while inside of a “duty” which counts for dungeons, guildhests, trials, and solo duties. Anyone messaging you will get a notice that they cannot receive your tell.
All of the dungeons are unlocked via various quest chains. Sastasha, Tam-Tara Deepcroft, and Copperbell Mines are unlocked through your story quests right after you get your Airship. Each adventurer's guild in each city state sends you to their respective dungeon before you can continue on with your story quests.
From there it's a mixed bag of quests outside of the dungeon (Halatali) and story quests (Totorak, Haukke, Brayflox). As you progress through the game you'll naturally unlock the dungeon when its time, assuming you're questing properly.
To enter a dungeon you'll need to use the Duty Finder. The Duty Finder generally requires you to have one tank, one healer, and two DPS to enter a dungeon. You can solo queue or queue with a group. No specific class has an advantage in the queue, except maybe Conjurer. Most classes, by virtue of just playing them (i.e. tanks build threat with their attacks), require little in the way of skill to progress through the dungeons, so there is little worry about grouping up with a lackluster group.
You are locked into the role for your class / job. If you're a Maruader, then you are a tank. If you're a lancer then you are DPS. There is no toggle otherwise.
Dungeons reward a fixed amount of XP on completion along with various treasure coffers (treasure chest). A treasure chest with an equipment drop will spawn after each boss while bonus treasure chests are available in out of the way locations which provide general loot drops and a chance at equipment dropping.
Gear drops on a need before greed system and goes by stats on the gear. Tanks need vit/defense, DPS needs str/dex/int, and healers need mind. The system is pretty good about picking out who needs what.
Set aside about 40 minutes for each dungeon, give or take ten minutes depending on the dungeon and how many times your group goes out of its way for treasure chests.
If you're DPS then don't be “that guy” who pulls everything with your bow because you want the dungeon to go faster but then get mad whenever you die, mob AI is very simple and they generally all pile onto the first person who pulls. Early on tanks have very little in the way of AoEs and responsive aggro management, so it's critical to let the tank pull.
Agree before the dungeon starts if you're going to chase after treasure chests or not. Again, another situation where “that guy” will drag mobs to the group in order to clear the way to a treasure chest (or they'll run through the mobs and aggro them to get the treasure chest). There is nothing wrong with hunting treasure, but make sure the group is on the same page.
If you're DPS and an enemy starts attacking you, switch to another one if available. That'll let the tank redraw aggro onto himself.
Always try to have Protect up on the group in some form. It really helps mitigate damage.
Red circles / triangles on the ground generally means the boss is charging up an attack. Don't stand in it! Likewise, as a tank, be sure to try and point bosses away from others, just in those scenarios where the group doesn't feel like moving.
Focus on healers first, always. They can make dungeons take forever if not dealt with quick enough.
Guildhests are designed, especially at the start, to train groups on different dungeon mechanics. You should play through them not only for the large XP reward, but for some strong pointers on different approaches to fighting enemies in FFXIV's dungeons.
Similar to dungeons, these face you off against one specific endboss type character. They usually have an instant wipe mechanic, so be on the lookout for that. Trials are unlocked through story quest progression.
After a group does enough dungeon, the option to do a limit break appears. Everyone has a “Limit Break” skill in their Actions & Traits menu. Activating it will increase defense (tanks), deal massive damage (DPS), or heal the group (healers). Coordinate with the group on when to use the limit break, which is a group wide bar.
There are various points in the game where a quest (primarily story quests) will have you enter a solo dungeon. These work the same way as a regular dungeon (no tells) except that they're for a single player only.
That's pretty much dungeons in FFXIV! Have fun exploring.