Step 1: What You Will Need
Forward current of the LED's
Voltage source higher than the LED's forward voltage
Multimeter (having two would be very useful if you are going to test many LED's)
Alligator clips or anything to hold on to the test leads
Solid wires (or leads from burned out components)
500 or 1000 ohm variable resistor
Step 2: Measuring the Forward Voltage
The forward current of the blue LED was 3.356 V at 19.5 mA. If you are powering it with 3.6 V, the resistor value to use is the next higher value for R=(3.6V-3.356V)/0.0195A)=12.5 ohm.
Measuring high power LED's with >350 mA forward current can be a bit tricky because when they heat up quickly, their forward voltage drops continuously. This means the current will be higher at a given voltage. To measure high power LED's, follow the same procedure and set the current. Quickly hold the value on both multimeters. It if's too late, you would have to let the LED's cool down to room temperature before taking the measurement again. For this reason, current limiting resistors are not used to power them. They must have a constant current source.
Don't know the forward current?
If the LED is part of an indicator light of a device, you can desolder one of the LED's lead and measure its current. This may not be necessary because you can measure its forward voltage across the leads while it is on.
Increase precision of current adjustment
I was unable to set it to 20 mA because the variable resistor had a large range of 50k ohm. You may use 500 ohm or 1k ohm. To allow coarse and fine adjustments, you may connect a higher and lower range variable resistor in series.