Aux boosts almost all emitter/deflector/sensor type abilities, and it also attributes to maneuverability. (i.e. turn speed) It's very important for most science abilities, and for cruisers who want to turn a bit better.
Is there a way to tell how much it actually boosts? Like if I change the power levels will the tool tips change to reflect the amount?
yes they do if you set it at 25 then change to a setting at 100 mouse over the ability and you can watch the numbers increase, if ther number arn't moving then that ability or item isn't being effected by that power stat
a note to all your cruiser pilot out there , if turning is getting you down try 65/60/25/50 you notice you can turn alot better and your tractors will be more effective which most crusier pilots will be using.
It also boosts the tachyon beam/shield drain ability. At 100 aux it will drain I think -85 from all shields per tick. If you use this ability at 100 it is very effective, if you use this ability when your aux is low, say after coming out of full impulse and near zero, it will barely do anything, like around -5 per tick.