Component with controllable resistance from Arduino?
The easiest way to drive an LED with variable brightness is to use one of the pulse width modulation (PWM) output pins on an Arduino.

A PWM pin takes an input value from 0 to 255 (or is it 0/1023? I don't remember.) and converts it to a ratio of on-time to off-time. At the highest value, the output pin is in the on state, 100% of the time. At the halfway point, the output pin is HIGH 50% of the time, and LOW 50% of the time.

Our eyes average out the flashes and see a variable brightness. It works very well.

If you absolutely must have a variable resistance controlled by a digital value, there are devices known as digital potentiometers. These are variable resistors controlled by digital inputs. For an Arduino, you should think about an I2C or SPI potentiometer, which is controlled by serial port commands. You tell it the resistance you want, and it sets itself to that resistance. 

As CrossRoads points out in their comment below, you still need a current limiting resistor to drive an LED from a logic pin, to protect both the LED and the output driver on the Arduino. You want to limit the current to the max current for your LED, and then use PWM to lower the brightness from the absolute max set by the current limiting resistor. CrossRoads explains how to calculate the resistor value.