First things first: the current power output of solar pannels IS the one currently "consumed" (I include heat losses and storage in battery if existant.)
So if you connect a 12 W LED that consumes 1 A at 12 V, and if your solar pannel's voltage-curent curve passes exactly at (12 V, 1 A) for the current amount of light, then your solar pannel will produce only 12 W.
If (in the same conditions) you decide to plug in a second (identical) LED, you will know have twice the load. If the solar panel were an ideal voltage source, you would now draw 2 A at 12 V (=24 V.) In practice, however, the voltage will be slightly lower (maybe 11.8 V), and at a lower voltage, the LEDs will also draw less current (maybe 0.9 A each), so you harvest 11.8*0.9=21.24 W.
If you continue adding load, you will reach the maximum power point (that is, have the load that gives the "best" trade-off between current and voltage, harwesting maximum power.) If you add yet more load, you will continue to increase the current, but the voltage will now decrease "too fast," making the output power decrease. Good solar battery chargers will make sure to always work at the maximum power point.
Measuring the current power output is simple (measure current and voltage at time t.) This is useful if you want to know what you are currently producing (for example if you store or sell the produced electricity.)
Measuring the "potential" power (i.e. the power at the maximal power point) is not possible directly (you cannot "just" add a sensor and know it.) There are however a few indirect solutions:
Add a variable load commanded by the Arduino, and from time to time, make it "sweep" through the range of possible values, while measuring voltage and current: the maximum voltage you measure is a good approximation of the maximal possible power (it is equal if you vary your load continuously.) In practice, you might use a few power-resistors switched on/off by relays commanded by the Arduino. If you choose the values of the resistors wisely, you can get many values by activating several values at once. If you already know approximatively the resistance you need to reach the maximum power-point in a low light condition, you can have one resistor for this value, and others to add to fill the "range" of possible values.
Add a battery and a MPPT (maximum power-point tracking) charger: as long as the battery is not full, you will always get the maximum power (so current current = maximal power), so you can just measure current power.
Measure maximum power "by hand" and calibrate. NB: this works only if your load is constant. So for example, on a bright day, with your LED connected, you have 12.2 V. When you measure "by hand" the maximal power, you get 120 W. On an average day, you get 12.0 V, and a maximal power of 80 W. On a slightly cloudy day, you get 11.8 V and 40 W. So you might approximate max power as 120 - 40*(12.2-voltage)/0.2. Of course, in reality, it will probably not be linear, so you will need some more points