Commissions are extra missions that you send your ships to participate in. For the duration of the mission those ships count as 'in combat' the same as your fleet when sortied on a chapter or event map.
On the commission page you will see a list of available commissions that you may select. These are mildly randomly generated and give various rewards however the time they take to complete varies between 20 minutes to 10 hours.
Once you select one you see an assortment of requirements as well as your currently chosen fleet for the commission. The first text tells you the required level of a number of ships: in this case it requires one ship to be at least level 10. The second text tells you how many ships you are required to select, in this case it is 4 or more. If you try to start a commission without a ship reaching the required level or selecting enough ships you will get an error message asking you to re organise the fleet/
The last is the viable ship classes you can chose from. In this example it is Destroyers, Heavy cruisers and Battlecruisers. You do not have to worry about the ship restrictions, as when you go to the screen to select your ships only those that you can send on the mission are selectable.
Time Limited Missions Edit
There are also Time limited commissions. While you battle across the enemy fleets in any sub-chapter you may get a notification telling you that you have obtained a new time limited commission
You can continue as normal or go back and to the commissions screen. These time limited missions will appear in the second tab.
These tend to have a variety of rewards at chance which do not appear in regular commissions. Such as skill books or even new ships.