The brightness of an LED is directly proportional to the current through the diode. When the LED conducts, the voltage accross it is always the forward voltage. The reason a resistor is used with an LED is to limit the current (and to develop the voltage drop needed in the series circuit.) So for a red LED, if you connect it to 12V, the resistor will have 10.15V accross it. The current accross this resistor would be V/R (Ohm's Law) and this will determine the brightness of the LED. You don't want to drive the LED more than its maximum current rating so you often have to determine a suitable resistor depending on the power supply voltage.
If we want to use the red LED with the mbed we have to figure out a safe resistor value. We know the supply voltage for the LED is 3.3V. The voltage accorss the resistor should be 1.45 (3.3-1.85). We can assume that we want 20mA for the LED (the current depends on the application of the LED and the LED itself.) If you are using the LED as an indicatior, you probably don't want it to blind you every time you look at it, so use less current. If we want a flashlight, we would probably drive the LED to its maximum current capability. So.. we know our resistor will have 1.45V across it and we want 20mA of current to flow, we can calculate the resistance needed with Ohm's Law: R = V/I -> R=1.45V/20mA = 72.5 ohms. You probably wont find a 72.5r resistor, so you can select 82 ohms or 100 ohms. See here for standard resistor values.
Anyway, I am not sure if you wanted to know all that (hopefully it can be of help to someone else). Your '12V' LEDs have a resistor in series somewhere as Hugo pointed out. You can try using them with 3.3V but they would probably be too dim. If you are daring enough, you can try to open the button to change the resistor. A seperate power supply is overkill for this application in my opinion..
I'd like to get my mbed board to drive two (possibly three) 12V LEDs through DigitalOut ports, one port for each LED. Obviously, I only get 3.3v out on those pins, and these drive normal LEDs fine.
The board is being powered by USB, so it would have to generate the 12V from the 5V USB input.
I'd prefer something to be *really* simple, as whilst I only wish to light 2-3 LEDs simultaneously, there will actually be lots of them which could be illuminated in different combinations.
No other IO is used, just DigitalIn, DigitalOut and the serial-over-USB to the host PC which powers the board.
Thanks