You can drive an LED with either a constant-current driver circuit or via a current-limiting resistor. The latter is an approximation to constant current drive, provided the supply voltage is sufficiently more than the forward voltage of the LED (5V is plenty). Since the eye is not very good at judging brightness to better than 30% or so, the resistor value is not too critical, it just has to be chosen to prevent the LED getting too much current in the worst case.
Different colour LEDs have very different forward voltages, note - you wouldn't usually share a resistor between different colour LEDs even if only one-at-a-time will be on.
A tricky situation is powering blue and white LEDs from 3.3V since the forward voltage of blue/white LEDs is about 3 to 3.3V and it varies somewhat with temperature. Luckily most Arduinos are 5V.
You calculate the voltage across the resistor = Vsupply - Vled, then choose the resistance value from R = V/I (I is the LED current you want).