Here is a list of the dungeons and how to gain entry:

Sastasha: Story Mission
Copperbell Mines: Story Mission
Tamtara: Story Mission
Halatali: Nedrick Iron-something (Vesper Bay)
Thousand Maws of Totorak: Story Mission (Into the Beast's Maw lvl 24)
Haukke Manor: Story Mission (Skeleton's in Her Closet lvl 28)
Brayflox Longstop: Story Mission (The Things We Do For Cheese lvl 32)
Sunken Temple of Qarn: Nedrick Iron-something (Vesper Bay)
Cutter's Cry: Silbold (Ul'dah Market Wards)
The StoneVigil: Story Mission (In Pursuit of the Past lvl 41)
Dzameal Stronghold: Carrilaut (The Observatorium)
The Wanderers Palace: Allene (Vesper Bay)
The Aurum Vale: Nedrick Iron-something (Vesper Bay)
Amdapor Keep: Nedrick in Vesper Bay (Must first complete AV before he gives you this quest)
Castrum Meridium: Story Mission
The Preteorium: Story Mission


Gotten from here: http://ffxivrealm.com/threads/dungeon-quest-thread-temp-sticky.5184/

Thanks Tann. Just what I was about to search for.