Washing hands, arms, face and feet etc. in a prescribed manner, before offering Salah (Prayers) is called Wudu (Ablutions).

Wudu literally means a bright, shining face.

Although the word Wudu (Ablutions) is not used in the Holy Qur-an; method of performing it before Salah (Prayers) is detailed in it and is commanded by Allah (God).

O you who believe! When you intend to offer prayer, wash your faces and your hands (forearms) up to the elbows, rub (by passing wet hands over) your heads, and (wash) your feet up to the ankles. If you are in a state of major impurity, bathe your whole body. But if you are ill or on a journey, or one of you comes from a nature call, or you have been in contact with women, and you find no water then take for yourself clean sand or earth, and rub with it your face and hands. God does not wish to place you in difficulty, but to make you clean, and complete His favor to you, that you may be grateful. Surah al-Ma"idah (The Table Spread With Food) Qur-an 5:6

The Messenger of Allah, Muhammad S talked specifically about the importance of Wudu (Ablution):

Abu Hurayrah R narrated that the Messenger of Allah S said "Allah does not accept the prayers of one who lost their Wudu (Ablution) until he performs it again."
Bukhari, Muslim, Abu Dawud, Tirmidi

There are many Ahadith (Sayings of Muhammad S) telling us that besides being a prerequisite for Salah (Prayers), Wudu (Ablution) has countless virtues. The Messenger of Allah, Muhammad S has also specified the rewards of doing Wudu (Ablution).

Muhammad S said: ‘If someone perform Wudu (Ablution) nicely and recite the following:

Ashhadu an la ilaha illahu wa ashhadu anna Muhammadan ‘abduhu wa rasuluhu

I bear witness that no one other than Allah (God) is worthy of worship and Muhammad S is His servant and Messenger, all eight doors of Jannah (Paradise) would be opened for him…’
Sahih Muslim

Prerequisite of Wudu (Ablution) is Taharah (Purification). For a valid Wudu (Ablution), one must be in a state of Taharah (Purification) and no impurity on the body or clothes. Wudu (Ablution) is a prerequisite of Salah (Prayers), the most important ‘Ibadah (Act of worship) in Islam. Part of an Ibadah (Act of worship) is also ‘Ibadah (Act of worship).

Doing Wudu (Ablution) is also an act of worship, and is a preparation to an important obligation, Salah (Prayers). It is recommended to be always in a state of Wudu (Ablution), but it is not mandatory.

Wudu (Ablution) can be made anytime, and it is not necessary to wait until hearing the Adhan (Call for Prayers).  

Jabir bin ‘Abdullah R Narrated that the Messenger of Allah S said, "The key to Paradise is prayer and the key to prayer is cleanliness (Ablution).
Tirmidi

Like all religious acts in Islam, Wudu (Ablution) also begins with a Niyah (Intention).

Niyah (Intention) is necessary before each Wudu (Ablution). It is not to be said loudly nor should it be articulated in specific words or sentences, just thinking about doing Wudu (Ablution) for a particular Salah (Prayers) is enough.

The Messenger of Allah, Muhammad S used to say a Du’a (Supplication) before Wudu (Ablution).

Allahummaghfirli dhanbi wa wassa’ li fi dari wa barik li fi rizqi

O Allah! Forgive my sins and bless my house and my livelihood
Nisai

The Messenger of Allah, Muhammad S did his Wudu (Ablution) in the following steps:

1. Say Bismillah [(I begin) With the name of Allah (God)].
2. Wash both hands.
   Use left hand to wash right hand, up to and including the wrist.
   Then, use right hand to wash left hand, up to and including the wrist.
3. Rinse mouth.
   Take water in cupped right hand, put in mouth, and then spit it out.
4. Take water in nose.
   Take water in cupped right hand; sniff lightly inside nose.
5. Wash face.
   Take water in both hands and wash face from forehead to the bottom of chin.
6. Wash forearms.
   Take water in cupped left hand, wipe right forearm with it, all the way up to the elbow, including it.
   Take water in cupped right hand, wipe left forearm with it, all the way up to the elbow, including it.
7. Wipe water over head/hair once.
   Wiping is done over complete head.

'Abdullah bin Zayd R reported that the Messenger of Allah, Muhammad S wiped his entire head with his hands. He started with the front of his head, then moved to the back, and then returned his hands to the front.

The Messenger of Allah, Muhammad S also wiped over his turban.

'Amr bin Umayyah R said "I saw the Messenger of Allah S wipe over his turban and shoes."
Bukhari, Ibn Majah

Bilal R reported that the Messenger of Allah S said, "Wipe over your shoes and head covering."
Ahmad

8. Wipe the inside and back of ears once.
   Wipe inside of both ears with index fingers at the same time
9. Wash both feet.
   Wash right foot including the ankle, with left hand.
   Wipe between each toe on right foot, with the small finger of left hand.
   Wash left foot including the ankle, with right hand.
   Wipe between each toe on left foot, small finger of right hand.

The Messenger of Allah, Muhammad S used to recite a Du’a (Supplication) after finishing his Wudu (Ablution).

Ashshadu an la ilaha illallahu wahdahu la sharika lahu wa ashshadu anna Muhammadan ’abduhu wa rasuluhu.

I testify that there is no deity worthy of worship but Allah (God) alone without any partners and I testify that Muhammad S is his slave and (last) Messenger.
Muslim