Ok... let's say you are trying to use this tricolor common Anode LED with your Arduino.
So you check the Data Sheet and find the Vfwd for each color as follows: Red, 2v. Green, 3.2v. Blue, 3.4v.
The basic circuit is going to be like this:
+5V > LED common Anode, each LED Cathode > each resistor > each Arduino pin
And you will want the Arduino pin to go LOW to turn the LED ON (5v is the Source and the Arduino pin is the Sink, so a HIGH pin (+5V) will result in no current flow and a LOW pin is like the pin is grounded, so current flows.).
And you want, say, 10mA current through each separate color of the LED.
So you use Ohm's Law to calculate the resistor value for each Cathode. R=V/I so
(Vsupply - VfwdLED)/0.010 = (5 - 2)/0.010 = 300 ohms for RED LED
(5-3.2)/0.010 = 180 ohms for GREEN LED
(5-3.4)/0.010 = 160 ohms for BLUE LED
So these are the three resistor values to start with, to give 10 mA through each color LED. But does this result in "equal brightness" as perceived by the eyeball? So you may need to adjust the values to give equal perceptual brightness.
Now, if you use three of the Arduino's PWM pins for the LEDs, you can use analogWrite() statements to control the LED brightnesses by PWM and code, as in the Examples>Basic>FADE example sketch in the Arduino IDE. So you can essentially arrive at almost any final color from the 3-color LEDs in combination at different brightnesses. The UNO's PWM-capable pins are 3,5,6,9,10,11.
I put my 3-color LED inside a ping-pong ball, with its individual cathode resistors connected to PWM pins of the Arduino, and my sketch fades each LED at different rates, so the ping-pong ball fades through "all" colors. Makes a nice "mood light".