The Altar Cave is the opening dungeon in Final Fantasy III. Located north of Ur it is, as the name of its musical track implies, the cave where the Wind Crystal rests.
A game's opening can give you an impression of the tone that it will take. The original Final Fantasy opens with the party directly outside of Cornelia, cutting directly to gameplay. Final Fantasy II opens into a battle against imprial troops which the party has no chance of winning, although some versions include a more formal opening to properly set the scene
Final Fantasy III begins with the protagonists falling down a hole into the cave. They quickly start bickering before realizing that the cave is crawling with monsters. This sets the tone for the game quite well.
As a first dungeon, the Altar Cave doesn't through anything particularly crazy at the player. There are random encounters, but if the party gets low on health, there is also a spring that will restore them to full health.
At the end of the dungeon is the crystal room, where the Onion Knights encounter the Land Turtle. Luckily, the cave also has a couple of SouthWinds that hit with a strong ice attack: The Land Turtle's weakness. Upon defeating it, they encounter the crystal which dubs them "Warriors of the Light". They then leave via the teleporter at the back.
Something that's easy to overlook is the early rooms of the Altar Cave: Since the party begins by falling down a hole, it may not occur to the player to go back in, missing out on some of the treasure that they could otherwise get. The hole itself is also still there, meaning you can go back to where the whole thing started, though there's no reason to.
The remake makes no changes to the cave layout itself, but makes two key differences. Gameplaywise, Luneth is the only character present in the opening sequence. Thus, the sequence is more dangerous, which sets the tone for the remake. Meanwhile the crystal chamber has been gussied up, making it more distinct compared to the other shrines.
The Mythril Mines are a mini-dungeon in Kazus, accessible in the town itself. The mine initially appears to be short, but a switch can be found, leading to a back area.