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Speed is a numerical indicator of how much time must elapse between creature actions, with higher numbers indicating longer delays (and therefore slower actions). All creatures have a default speed of 900; this value can be overridden by a [SPEED:#] creature token in the creature raw file. Note this internal value for SPEED is not the same as the "Speed" stat shown on the screen in Adventure mode.
The way it works is that the game doesn't let most creatures act every single frame - an "average" creature acts only once every 10 frames. In order to achieve this, there is a delay on every action that a creature takes.
Basically, the hundreds digit and higher represent whole turns that are skipped just filling up the waiting counter until it reaches the point of their "speed". "Speed" is a bit of a misnomer, actually, as it's really more "turn delay".
This is why most creatures have 900 speed - they are delayed 9 turns before they get their 1 turn of action, and have to wait for their delay counter to fill up before taking their next action.
A creature with 975 speed, meanwhile, (which is functionally where most dwarves fall, thanks to low agility) has to wait 9 turns, then rolls a random chance to see if it goes that turn or not - with a 75% chance of having to wait another turn.
Agility acts as a multiplier on speed - if someone had only 400 speed/delay to begin with, and another creature had 900 speed/delay, then regardless of whether it was low or high agility, as long as they had the same amount of agility (and equipment), the 400 speed/delay creature would act twice as fast. A creature with 0 speed/delay has no benefit or penalty from agility at all. -- NW_Kohaku 
400 is the fastest speed that you will see in the wild without modding; creatures this fast will move at twice the speed of a typical dwarf.
(As mentioned above speed is used in a different way in Adventurer mode)
Your adventurer's speed stat is displayed at the bottom of the screen; higher values mean that the adventurer moves, acts and attacks faster. The speed stat is calculated by speed_stat = floor(1,000,000/(SPEED + 100)), with a maximum value of 10,000.
In addition to movement, many non-combat actions are also affected by speed, including eating. It is usually in an Adventurer's interest to have as high a speed as possible to allow for faster movement and more attacks relative to an enemy. An adventurer with low speed will notice both friendly and enemy units taking multiple turns for every turn they take. High speed may also be beneficial for those with lag issues - a higher speed means less AI turns being calculated between each player move, decreasing the processing delay between player actions.
Many factors affect effective speed. A high agility attribute results in faster actions. Higher strength also increases speed, but because it also increases body mass (more weight = slower) the effect is not as pronounced as agility. Encumbrance (carried items) lowers speed relative to the total weight. Lying down, falling down, or being stunned approximately halves a creature's speed. Drowsiness, hunger, thirst, disabilities, and dizziness can also decrease effective speed.
Some actions also impose speed modifiers, such as wearing armor, moving quietly, running, and swimming.
Skills can reduce the impact of some modifiers on speed.
- Training in Armor User reduces the speed penalty for wearing armor.
- Training in Ambusher reduces the penalty for hunting and sneaking.
- Training Crutch-walker (and carrying a crutch) reduces the penalty for a disabled leg.
- Training Swimmer improves speed while swimming; a high-level swimmer may actually move faster in water than on land.