Intention. Since Islam teaches us that intention forms the basis of any action, it is necessary that, before you perform Wudu, you make the niyyat or intention for it. This will not be a verbal act - you only have to mentally prepare yourself for an act that will cleanse and purify you.
Say Bismillah. Recite bismi-llahi ar-rahmani ar-rahim (In the name of Allah, the most Gracious, the most Merciful).
Begin by washing your hands. Completely wash your right hand three times with clean water, then move on to the left and repeat. When washing, remember not to miss any part of the hands - wash between the fingers and thumb, too.
Rinse mouth. Take water in your right palm and rinse your mouth with it, thrice. Do not swallow the water on purpose, but spit it out every time.
Cleaning the nose. Using your right hand, carefully and gently put water into your nostrils, before exhaling it. Perform this step three times.
Wash your face. Thoroughly wash your face, beginning from the hairline on your forehead and from one ear to the other. This step, too, must be performed three times.
Wash your arms. Begin with your right hand. Wet it with water, and wash from your wrist up to your elbow. Repeat thrice; then do the same to your left arm.
Perform Masah. Usually, Masah refers to the religious act of cleaning one's head, in a specific way, with a small amount of water. For performing Masah, wet your hands and place your palms, flat, on the top of your head (where the hairline begins). Wipe them to the back of your neck, and then back again to the front. Now wet the back of both ears by placing your thumbs behind them. Use your index and middle fingers to wash the front and back of your ears. Masah is done only once.
Wash feet. Start with your right foot. Wash it thrice from the toes, up to the ankles. Make sure you wash between your toes, too. Do the same to your left foot.
Remember: Once Wudu is performed, it does not need to (but can) be performed again until it is invalidated.