The dungeon level is displayed to the far left of the status line. You start the game on dungeon level 1, where it might read:
Dlvl:1 $:1699 HP:13(13) Pw:5(5) AC:8 Xp:1/0 T:1
The type and difficulty of monsters generated are based on the dungeon level.
Dungeon levels and difficulty levels
Some game mechanics are affected by the current level difficulty. Normally this is the same as the level's physical depth number. If a branch builds up (like Sokoban), level difficulty increases with decreasing depth. However, if you are carrying the Amulet, the level difficulty is always the depth of the deepest level you have reached (i.e. the Sanctum). Furthermore, if you are in the Planes, this figure is increased by half your experience level (rounded down).
Special dungeon levels
Go to Nowhere. Are you sure? [ynq] (q)
If you decide to answer yes, you die:
You scream in agony as your body begins to warp... You cease to exist. Your possessions land on the floor with a thud.
An energized cloud of dust begins to coalesce. Your body rematerializes, and you gather up all your possessions.
Dungeon levels -1 to -5 are the elemental planes and the astral plane. These are not accessible via teleportation, so the numbers are only useful in wizard mode. If you attempt to teleport to any level between -1 and -9, you find yourself high above the clouds, and plummet to your death:
You are now high above the clouds... Unfortunately, you don't know how to fly. You plummet a few thousand feet to your death.
You are now high above the clouds... You fly down to the ground.
Similarly, if you are levitating, you will also land safely and escape the dungeon:
You are now high above the clouds... You float gently down to earth.
Teleporting to level -9 additionally gives the message:
You feel deliriously happy. (In fact, you're on Cloud 9!)
Teleporting to level -10 or higher sends you to heaven:
You arrive in heaven. "Thou art early, but we'll admit thee."
Your death is then recorded as "went to heaven prematurely".
If an amulet of life saving rescues you from teleportation to any negative dungeon level, you will "find yourself back on the surface" and escape the dungeon.
Mazes of Menace - outlining the list of levels in NetHack.