Wudu is the washing done before prayer. It takes two or three minutes at a tap, and it lasts until something breaks it, so one washing often covers several prayers.
What the Qur’an requires
“O ye who believe! when ye prepare for prayer, wash your faces, and your hands (and arms) to the elbows; Rub your heads (with water); and (wash) your feet to the ankles.”
Those four are the core, and every school holds them obligatory. The rest of the steps below are how the Prophet Muhammad (ﷺ) did it, and are followed as his practice. The four from the verse are marked.
Step by step
Niyyahintention
Intend in your heart that you are washing for prayer. Nothing needs to be said aloud.
Bismillahin the name of Allah
Say "Bismillah" as you begin.
Hands
Wash both hands up to the wrists, three times, running water between the fingers.
Madmadahrinsing the mouth
Take water into the mouth with the right hand, swirl it round and spit it out, three times.
Istinshaqrinsing the nose
Draw a little water into the nose with the right hand and blow it out with the left, three times.
Face
Wash the whole face, from the hairline to the chin and from ear to ear, three times.
in the Qur'anArms
Wash the right arm from the fingertips up to and including the elbow, three times, then the left.
in the Qur'anMasahwiping the head
With wet hands, wipe over the head from the front to the back and back again, once.
in the Qur'anEars
Wipe the inside of the ears with the index fingers and the back with the thumbs, once.
Feet
Wash the right foot up to and including the ankle, three times, making sure water reaches between the toes, then the left.
in the Qur'an
Right side before left throughout. Washing each part three times is the Prophet’s (ﷺ) practice; once, done properly, is valid. Don’t waste water.
Where the schools of Islamic law differ
The schools of Islamic law differ on some details, for example whether the intention, the order of the steps, and rinsing the mouth and nose are obligatory or strongly recommended, and how much of the head must be wiped.
What breaks wudu
Agreed by the schools: using the toilet (urine or faeces), passing wind, deep sleep, and losing consciousness. After any of these, wash again before the next prayer.
Where the schools of Islamic law differ
Other things are disputed. For example, the Hanafi and Hanbali schools count flowing blood and heavy vomiting; the Maliki, Shafi’i and Hanbali schools count touching the private parts directly, which the Hanafis do not. If you are unsure whether your wudu has been broken, most of the schools say you go by what you are certain of, so a doubt alone does not break it.
Some things need a full bath, ghusl (ritual bath), rather than wudu: after sexual intercourse or emission, and after menstruation or bleeding following childbirth ends.
If there is no water
If you cannot find water, or using it would harm you through illness, the same verse allows tayammum (dry ablution): strike clean earth or dust lightly with the palms, then wipe the face and the hands. The Hanafi and Shafi’i schools wipe the arms to the elbows as well, with a second strike.
“and ye find no water, then take for yourselves clean sand or earth, and rub therewith your faces and hands, Allah doth not wish to place you in a difficulty”