Combat Speed is a largely self-explanatory unit attribute; it's how fast a unit is in combat, specifically for movement speed. This is not Map Movement, which measures how quickly a unit can travel in extended periods.

According to the manual, a unit's Combat Speed is exactly how many squares they can cross in a single combat round. Testing in Dominions 5 revealed that a point of Combat Speed equated to roughly 3/4 of a square, but the battlefield was significantly smaller in that game than it is now.

Units get their rounds asynchronously, with a random order that is not well understood, and a delay between them that is even more esoteric. In the first four games, every unit took their turn one-at-a-time, and the shift to simultaneous movement occurred in Dominions 5. It took a while for Illwinter to tune certain things around this mechanic, particularly how fellows aim their missiles.

Typically, units cannot enter a square that doesn't have enough free Size points for them. They can move diagonally around others, but this takes 1.5 Combat Speed points instead of 1. Alternatively, a unit 3 Size points bigger than another unit in the square can displace them; this shoves the displaced unit into a random adjacent square, possibly the square the displacer came from, but this takes 1 extra Combat Speed point. Displacing an immobile "scenery prop", such as a Bush, instantly flattens and destroys them.

Displacing an enemy is only possible for Tramplers, and all displacement done by a Trampler causes damage. Combat Speed does not affect the damage output, but it does affect the "damage-per-second"; units who use up all their Combat Speed can't move again until their next round comes up, after all. Trampling also caused the Tramplers a bit of a delay in Dominions 5, which was reduced by both Combat Speed (their speed with moving) and Size, but it's not clear if that still applies.

Mounted units make use of their high Combat Speeds to make Charge Attacks, getting a damage bonus that scales with that and their Size.

Last-but-not-least, the Flying don't really care about Combat Speed, for all that we know about the game.

Leg Afflictions can reduce Combat Speed. Having a Limp will halve your Combat Speed, for example, while being Crippled will reduce it to 2. If you have both, the Cripple overrides the Limp.

Swiftness is the most straightforward Combat Speed bonus, providing a multiplier. The Air Bless provides stackable chunks of 30%, while the spells and items provide 100%.

Last but not least, Quickness effectively doubles your Combat Speed by doubling the rate of your rounds, while Slowness halves it.

Combat Speed is highly dependent on the Size of the unit and is somewhat impacted by their Armor. To be specific, a unit's Encumbrance from Armor and Shields is subtracted from their Combat Speed.

Yes, immobile units still have Combat Speed. This is possibly a holdover from the games before Dominions 5; it isn't clear if it affects the random order of rounds.