what it is
The Dungeon is one sprawling brick structure generated on a single side of every world, always the side opposite the jungle and the same side as the snow biome unless a drunk world is involved. It opens at a castle-shaped building on the surface and runs down to just short of the Underworld, with spawn rates climbing the deeper you go.
Three things are rolled at world generation: the brick color, the background wall patterns, and which of three entrance designs you get. The first and last are cosmetic. The wall patterns are not, and they decide which enemies you fight here in the back half of the game.
getting in
The Old Man stands at the entrance. Speak to him at night and a Curse option appears, which turns him into Skeletron. During the day he will not offer it. Beating Skeletron opens the Dungeon and unlocks the Clothier, who moves in as soon as a house is free. The Old Man does not come back afterward, and a Clothier Voodoo Doll is the only way to fight Skeletron again. The Mechanic is bound somewhere in the depths and joins you when you talk to her, which is most people's reason for doing a full sweep on the first visit.
the guardian line
You can enter the Dungeon before killing Skeletron, but barely. Dropping below minus three feet inside it summons a Dungeon Guardian, which will kill nearly any character in a single hit and cannot be fought in any practical sense. Killing one does not grant access either. The only thing it drops is the Bone Key.
Two signals tell you the line is near: the music switches to the Dungeon track, and a Depth Meter starts reading Level or a count of feet underground. Entrances generate at varying heights but always some way above zero depth, so the entrance building and a screen or two below it are safe to poke around in. Chests that happen to generate above the surface line can be opened freely, although they never hold the unique items. One edge case: standing outside the Dungeon while touching its exterior wall, below minus three depth, can spawn a Guardian anyway.
the chest order
Each Dungeon chest holds one unique item, and those items are handed out in a fixed sequence rather than rolled at random. The Muramasa comes first, then the Cobalt Shield, then the Aqua Scepter, the Blue Moon, the Magic Missile, Valor, a Golden Key and the Handgun.
Because the entrance is a single unbranched corridor, the first few are predictable. If the first chest sits above minus one hundred feet it will be a wooden one holding a Golden Key; deeper than that and it holds the Muramasa instead. The next along holds the Cobalt Shield, and so on down the list. The prediction falls apart once the corridor splits, which it does quickly. The Shadow Key and the Ram Rune generate alongside another unique item rather than taking a slot, so they never shift the sequence.
cracked bricks and what is behind them
Some passages are sealed with Cracked Dungeon Bricks. They look like ordinary Dungeon Brick with a slightly crumbled texture and they drop nothing when broken. Hitting one can chain-break the ones beside it, the same way Desert Fossil collapses. They also give way on their own when you walk on them, hook onto them, or clip them with a stray projectile.
Behind them is usually a large rectangular room lined with spikes, sometimes flooded on top of that. A Dangersense Potion highlights the cracked blocks and is worth drinking before you open any walls. Those flooded spike rooms are also the best fishing spots in the Dungeon, if you can get into one without landing on the floor.
after Plantera
When Plantera dies, a message about screams echoing from the Dungeon appears and the enemy roster is replaced by a much harder one carrying much better drops. The Biome Chests become openable at the same time.
Which of the new enemies you get is decided per area by the background wall behind it. A Dungeon can hold several wall types in different sections or only one, and there is no way to tell from outside; you have to walk it. The decorative faction banners hung on the walls are the tell, so a Diabolic Sigil in a room means Diabolists spawn there. The pairing is a strong tendency rather than a hard rule: a Skeleton Sniper can still appear in a plain brick wall area, and higher luck raises the chance of off-list spawns.
One structural detail is worth remembering. Most Dungeons have plain brick walls hidden behind their foreground blocks regardless of what the visible walls look like. Mine those blocks out after Plantera and you have built a brick wall spawn zone wherever you dug. Bone farming also gets worse from here, since the new spawns crowd out the enemies that drop them and drop none themselves.
after Golem
Once Golem and Skeletron are both dead, four Cultists appear at the Dungeon entrance. Killing all four summons the Lunatic Cultist, and killing him starts the Lunar Events, which run straight into the endgame. If you have come back only to loot, fish or farm, walk around them.