This is a nice little list of dungeon levels/requirements that was originally posted on Blue Gartr by Ratatapa. I edited some of the areas to be more specific.
Sastasha -- Western La Noscea (Lv. 15)
How to Unlock : Story
Tam-Tara Deepcroft -- Central Shroud (Lv. 16)
How to Unlock : Story
Copperbell Mines -- Western Thanalan (Lv. 17)
How to Unlock : Story
Halatali -- Eastern Thanalan (Lv. 20)
How to Unlock : Talk to the NPC outside of the Walking Sands in Vesper Bay
The Thousand Maws of Totorak -- South Shroud (Lv. 24)
How to Unlock : Story
Haukke's Manor -- Central Shroud (Lv. 28)
How to Unlock : Story
Brayflox's Longstop -- Eastern La Noscea (Lv. 32)
How to Unlock : Story
The Sunken Temple of Qarn -- Southern Thanalan (Lv. 35)
How to Unlock : Talk to Nedrick Ironheart outside of the Walking Sands in Vesper Bay
Cutter's Cry -- Central Thanalan (Lv. 35-38)
How to Unlock : Talk to Silbold in Ul'dah Market Wards
The Stone Vigil (Lv. 41-44)
How to Unlock : Story
Dzemael Darkhold -- Coerthas (Lv. 41-44)
How to unlock : Level 41 - quest by your grandcompany (GC), should be at left or right side. The observatory in Coerthas south of Dragonbone by Mor'Dhona entrance
The Wanderer's Palace (Lv. 50)
How to Unlock : Talk to the NPC near Waking Sands / Allene (Vesper Bay)
The Aurum Vale -- Coerthas (Lv. 47-50)
How to Unlock : Talk to Nedrick Ironheart outside of Waking Sands