In general terms the amount of light determines the maximum amount of power the panel can deliver.
You'll often find 3 things quoted:
1) max power
2) open circuit voltage
3) short circuit current
The maximum power is pretty much never achieved. The panel needs to be pointed directly at the sun in the middle of the day with no cloud or haze. Practically, it's not going to happen too often. The max power also falls as the cells age or as dust settles on the glass over the panel.
The open circuit voltage (say 17.5 volts for a nominally 12 volt panel) tells you the maximum voltage that the panel will deliver. It's of limited value because this is at zero current, i.e. zero power.
The short circuit current indicates the maximum current a panel can deliver. It is also of limited value as into a short circuit the panel is also delivering no power.
Multiplying the open circuit voltage by the short circuit current gives a figure in watts that means nothing and can never be achieved by this panel -- so don't do it
The power you can get from a panel depends on the current you draw from it, and the voltage that appears across the panel when you do so. The power varies with load, and the "best" point (i.e. maximum power) varies with the amount of illumination on the panel.
There are power tracking regulators that continually hunt to find the optimum point, which can improve efficiency of (say) battery charging by 10 to 15%.
So the answer, in short, is yes, yes, and both (for a given current)