Speed is both a monster property in NetHack (also referred to internally as movement rate), and an intrinsic/extrinsic related to that property.
Property
For every turn that passes a monster will gain its movement rate in "movement points"; if the monster has 12 or more movement points (defined as NORMAL_SPEED[1]) in a turn it gets a move that turn (and more than one if it has another 12 points remaning after this, etc.) Thus a monster with a speed of 12 gets one move per turn, a monster with a speed of 1 gets one move every 12 turns, and a monster with a speed of 18 gets 3 moves every 2 turns.
Being encumbered reduces the player's effective movement rate after adjustments. A burdened player has three quarters their normal movement rate, a stressed player one half, a strained player one quarter, and an overtaxed player one eighth[2].
Multi-turn actions (e.g. entering in "10s" to search ten times) consume one move per command issued plus whatever additional time is required for these moves. In contrast, certain actions will render the player immobile for a certain amount of turns; during those turns the player's movement points are simply drained as though the player was executing moves during that time, but without actually allowing the player an action. Fainting, being paralysed, engraving a long sentence, etc. are examples of actions that cause this.
Intrinsic
A player with the speed extrinsic (obtained through speed boots, a potion of speed, or the haste self spell), also known as a very fast player, grants a guaranteed six extra movement points per turn, and one turn out of three another six extra movement points[3], thus having the player move with an effective movement rate of 20, or five thirds normal, giving five moves every three turns. In the absence of extrinsic speed, a player with intrinsic speed (obtained through wands of speed monster, quantum mechanic corpses, etc.), otherwise known as a fast player, grants six movement points two turns out of three[4], resulting in an effective movement rate of 16, or four thirds normal, giving four moves every three turns.
The extrinsic overrides the intrinsic
A monster with intrinsic speed (gained by e.g. a wand of speed monster) has four thirds their normal movement rate[5]; a slowed monster (by means of a wand of slow monster) has two thirds their normal movement rate[6].