I checked out all tooltips in the game and their official website. Since FTL did not come with a manual I have no clue here. I know that they have something to do with recharge rates and percentages but I can't find anything specific about it.
It varies from system to system as follows ("manned" or 0 level/1 level/2 level)
- Engines - increases evasion 5%/7%/10%.
- Shields - increases recharge rate 10%/20%/29%
- Weapons - reduced charge time 10%/15%/19%
- Piloting - increase evasion 5%/7%/10%. Also, with no pilot and no auto-pilot you get 0% evasion regardless of the level/power of engines!
- Melee - increases combat damage, 0/10%/20%
- Repair - increases repair speed 0/10%?/20%?? (seems likely)
Source here, and also from help in the comments.