Z-Levels are opened up when building a down staircase from your dungeon.
Note that the map is 3D and has a vertical axis, which can be viewed one 2D cross section at a time.
You may only build one down staircase and that staircase will link to the middle of the next z-level. Portals do not work across z-levels. In this regard, z-levels are not done in the same way as certain other games (including Dwarf Fortress).
The current vanilla configuration is found here:
This is the current vanilla configuration shown as a table.
|Inhabitants||Starts at Depth||Ends at depth||Attack chance||Landscape|
|Water Creatures||7||10||None||Underground sea|
|Lava Creatures||10||Never||None||Underground magma sea|
|Dark Elves (for lawful keepers)||5||Never||60%||Resources\river\lava|
|Dwarves (for evil keepers)||5||Never||60%||Resources\river\lava|
You can keep digging down forever.