Z-Level describes vertical space (depth or altitude) within Dwarf Fortress, analogous to the Z-axis in geometry which extends out of the page towards the viewer. Each layer of view is a discrete z-level with a value relative to the bottom of the map space, indicated in the lower right corner of the screen. The player moves their view from one z-level to another by using to move up and to move down.
In v0.31 the number of available z-levels has ballooned dramatically from prior versions with the addition of the Underground cavern systems, from a few dozen to potentially hundreds. The default settings produce levels with around 150 z-levels of land (for a embark with average elevation changes) with an additional 15 z-levels of empty sky space above the highest point of land.
Numerous factors available in world generation impact the available z-levels, and can alter the depth of the map to a minimum of 6 and a maximum well in excess of 600. (Worldgen has 400 z-levels; maybe one can force 200 levels of sky.)
Reducing the number of z-levels, especially cavern levels, can reduce processor demand and boost framerate.
There is another Z-level display in the upper right corner of the main view. This displays the player z-level viewpoint relative to the surface z-level.