There are four difficulties consisting of Normal (50-60), Heroic (60), Mythic (60), and Mythic+ (60).
There are 8 "traditional" dungeons in Shadowlands. Of these dungeons, 4 are available for leveling players, with another 4 available at max level.
The item level of dropped gear depends on the difficulty of the dungeon and Shadowlands will not contain Warforging or Titanforging.
Normal Dungeons drops Item Level 157 gear at Level 60.
Heroic Dungeons drops Item Level 170 gear.
Mythic Dungeons drops Item Level 183 gear.
To queue for Heroic Dungeons through the Dungeon Finder, you must be a certain item level (157), but you can walk in with a premade group at any point.
Torghast is a different type of dungeon. It can only be unlocked at level 60, following a short questline, and offers an ever changing dungeon for between 1 and 5 players. The further you get through it, the greater the challenges.
The Necrotic Wake: 50+
Mists of Tirna Scithe: 55+
Halls of Atonement: 57+
Theater of Pain: 60
De Other Side: 60
Spires of Ascension: 60
Sanguine Depths: 60