Materials needed:

-10k potentiometer (variable resistor)
-Breadboard
-LED
-9V battery
-9V battery connector
-Wire
-1k resistor
-250 - 270 ohm resistor

Note: You can get all of these objects at RadioShack

Step 1: Putting Potentiometer in Place

-Take the red and black wire from the battery connector and put them on the side rails.
-Attach a wire to each lead from the potentiometer and place on breadboard
-Put the LED somewhere on the breadboard

Step 2: Putting Resistors in Place

-Put 1k resistor in the middle lead of the potentiometer and put the other end in a different rail
-Put the 270 ohm resistor in the longer lead of the LED and attach to same rail as 1k

Step 3: Wiring

-Attach the left end of the potentiometer to the positive of the battery
-Attach the right end of the potentiometer to the negative (ground) of the battery
-Attach the shorter lead of the LED to the negative on the battery

Step 4: Finished Product

Connect the battery and turn the potentiometer and the brightness of the LED should change. If it doesn't make sure you did everything correctly.
What this circuit basically does is that it increases and decreases the voltage the LED gets by changing the resistance. You can change the value of the resistors to make the LED brighter or dimmer.