The Mythril Mine is a small cave located to the west of the Chocobo Farm, through the swap, on the eastern continent. Traveling through to the other side takes you to the Junon Area.
The story takes you through the Mythril Mine just once but you can return to the are at any time. There are a few lines of dialogue between your team and the Turks when you arrive, but the Mythril Mine is fairly small and easy to pass through. Check out the Mythril Mine walkthrough page for more details.
There are a few items that you can pick up in this area including:Ether
Long Range Materia
The map below shows the location of the Mythril Mine on the World Map: