Power is related to voltage and current: P = V * I. You need to know 2 of the 3 to compute the other. If you have a resistive load and you know the resistance R, you additionally know (via Ohm's law) that V = I * R and various substitutions let you come up with P = V^2 / R or P = I^2 * R, but then you need to know R.
Note that many loads are NOT resistive. many loads are "constant power" - their current increases as the voltage decreases so that P stays roughly constant. Some loads behave somewhere in between, and some esoteric loads behave really weirdly (like fluorescent tubes).
If you just have a "black box" circuit, you have no idea how voltage will relate to current. It's possible to build a circuit that does just about anything.
If you have a multimeter, you can measure the current directly. You have to put the meter "in-line" with the circuit. That is, you unhook one supply lead, and hook the meter up between them on the current (amperage) setting. For AC loads, there also exist "clip-on" type meters, though their accuracy varies, and this doesn't work on DC, which it sounds like you've got.