Straight from the designers at the Daystrom Institute, the following is an explanation of Power Allocation.
Adding Power to Weapons - Increases energy weapon damage.
Adding Power to Shields - Increases shield regeneration.
Adding Power to Engines - Increases Starship Speed and Starship Defense (making you harder to hit)
Adding Power to Auxillary - Increases Turn Rate, Cloaking and Cloak Detection.
Most powers are also affected by power levels.
Power to Shields improves most shield powers
Power to Aux improves most science powers
We hope this will assist in your future missions Captains. Q speed and good luck.