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Damage Controls increase the repair speed of on-board repairs of battle damage (not malfunctions). A ship might be able to repair damaged systems within hours instead of limping to a shipyard. This costs twice as many MSP as repairing a malfunction. If, for example, your biggest component is an engine costing 800 MSP to fix if malfunctioning (see Max Repair), it would cost 1600 MSP to repair using Damage Control if that engine were destroyed in battle. If your ship doesn't have 1600 MSP, then it needs to be repaired in a shipyard.
DC value is the chance to repair a component within a given time. Usually that works out to some hours, although a very large number of modules could shorten this time considerably. However, note the very high MSP cost of emergency repairs.
You don't need a DC module for on-board repairs; some capabilities are provided by engineering spaces. A Damage Control module will increase your ship's Damage Control value by 10, while an Engineering Section adds only 1 to the DC value. Damage Control modules do not extend maintenance life.
Armor is not considered a ship component and still needs to be repaired in a shipyard.
To initiate repairs, go to the Damage Control tab of the Individual Ship window. Select a component and click "Begin DC", or "Add to Queue".
Word of God
The psuedo code is as follows:
Repair Chance = Round Down ((Increment Length in Seconds / Item Cost) x Highest Damage Control Rating)
If Repair Chance > 1000 Then
Repair Done = True
If Random Number(1000) <= Repair Chance Then Repair Done = True End
Highest Damage Control Rating is usually the damage control rating of the ship. However, if the ship is in a hangar and the mothership has a higher damage control rating, then the damage control rating of the mothership is used. This could be used by fighters on a carrier or it could be a huge repair ship that takes full size ships into its internal hangar. In this case, the maintenance supplies of the mothership are used.