Endgame, improved dungeons in Diablo 4
Nightmare Dungeons are for late-game players and are upgraded versions of the dungeons[1] you've already found, containing more dangerous monsters and challenging objectives:
"In the Nightmare Dungeon system, you can obtain keys or seals that allow you to upgrade a dungeon in the world to become a Nightmare dungeon, which has additional affixes and powers. The effects on the dungeon are based on the You're going to get a lot of keys. You might find keys that are less suited to your style of play and you don't have to do them. You have a lot of player choices.
Nightmare Dungeon Monsters in Diablo 4
Elite Monsters in a Nightmare Dungeon
Unlocked after finding the first Nightmare Seal
According to Blizzard, players can begin doing Nightmare Dungeons when they receive their first Nightmare Seal. It's unknown if this first Sigil will come from random loot, boss loot, or a reward (for, say, Whispers of the Dead). A Sigil is associated with a specific dungeon, so once found, a player can search for the dungeon named on the Sigil.
"Nightmare Dungeons in Diablo 4 are unlocked by finding your first Nightmare Seal. A Seal matches a certain dungeon somewhere in Sanctuary."
Found in Cleared Fortresses
Entrances to regular dungeons appear after defeating monsters in a stronghold. Since nightmare dungeons are upgraded versions of normal dungeons, we expect their locations to be the same (i.e. in cleared fortresses). It's not yet known if you can upgrade any dungeon to Nightmare or just select one. No matter what kind of dungeon you choose to do, you should always buy Diablo 4 items or gold to make your character more powerful.
Entrance to Nightmare Dungeon in Diablo 4
A player upgrades a normal dungeon to a nightmare dungeon using a nightmare sigil. Blizzard uses the word "key" interchangeably to describe them, so we believe they're identical to the Dungeon Key system detailed at BlizzCon 2019. In this system, keys have difficulty levels and affixes that modify the dungeon. Because you know the affixes ahead of time, you can choose to throw the key away or use it on a found dungeon, adapting your gear to better withstand the affix conditions.
Blizzard says Nightmare Sigils can be acquired with the new Whispers of the Dead endgame system, which provides routine tasks to do around Sanctuary for rewards:
"[The Tree of Whispers] gives you access to Nightmare Sigils, which take you into Nightmare Dungeons."
Write Joe Shely Diablo IV game director Twitter icon svg
There may be other ways to get Nightmare Sigils (e.g. rare loot drops), but Blizzard hasn't confirmed anything.