What Are the Pillars of Wudu?
Question

What are the essential parts (arkan, lit. “pillars”), obligatory (wajib) parts and sunnahs of wudu?
Praise be to Allah.Pillars of wudu
There are six essential pillars of wudu:

Washing the face, of which the mouth and nose are part.
Washing the arms up to the elbows.
Wiping the head.
Washing the feet up to the ankles.
Washing these parts of the body in the proper order.
Doing these actions consecutively (that is, washing one after the other, with no lengthy lapse of time in between).

Allah, may He be exalted, says (interpretation of the meaning):
“O you who believe! When you intend to offer As-Salah (the 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 ankles” [al-Maidah 5:6] (See ar-Rawd al-Murbi‘ ma‘a Hashiyat Ibn Qasim, 1/181-188)
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said:
“What is meant by the obligatory (fard) parts of wudu here is the essential parts (pillars).
Thus we may understand that the scholars (may Allah have mercy on them) use different expressions, so they may describe the obligatory (fard) parts as essential parts (arkan, lit. pillars) and vice versa.” (Ash-Sharh al-Mumti‘, 1/183)
We have mentioned previously that fard is the same as wajib [both mean obligatory] according to the majority of scholars. Please see question no. 217272 .
So the obligatory (wajib) parts of wudu are its pillars or essential parts, which may also be described as fard. They are the actions without which wudu cannot be valid.
Saying Bismillah before Wudu
With regard to saying Bismillah before doing wudu , Imam Ahmad was of the view that it is obligatory.
However, the majority of scholars are of the view that it is one of the Sunnahs of wudu and is not obligatory. 
Sunnahs of wudu
The sunnahs of wudu are numerous. Shaykh Salih al-Fawzan (may Allah preserve him) said:
The sunnahs of wudu are as follows:

Using siwak, which is to be done when rinsing the mouth, so that by means of the siwak and rinsing the mouth, the mouth will be cleansed for worship and be prepared for recitation of Quran and conversing with Allah, may He be glorified and exalted.
Washing the hands three times when starting wudu, before washing the face, because of the hadiths which speak of that, and because the hands are the tools used to transport water to the other parts of the body, so washing them is prudent action before proceeding with wudu.
Starting by rinsing the mouth and nose before washing the face, because starting with them is mentioned in the hadiths; and one should do them thoroughly when not fasting.

What is meant by washing the mouth thoroughly is circulating water to all parts of the mouth, and when rinsing the nose it means taking up water to the deepest part of the nose.

In the case of a thick beard, ensuring that water reaches throughout; and making sure that water gets in between the fingers and toes.
Starting on the right in the case of the hands/arms and feet, before the left.
Washing more than once, up to three times, when washing the face, hands/arms and feet.” (Al-Mulakhkhas al-Fiqhi, 1/44-45)
Another Sunnah is wiping the ears , according to the majority of scholars. Imam Ahmad was of the view that wiping them is obligatory. 

Du’a after wudu
After doing wudu, it is mustahabb to recite the following supplication : Ashhadu an la ilaha ill-Allah wahdahu la sharika lah, wa ashhadu anna Muhammadan ‘abduhu wa rasuluhu. Allahumm aj’alni min al-tawwabina waj’alni min al-mutatahhirina, subhanaka Allahumma wa bi hamdika, ashhadu an la ilaha ill anta, astaghfiruka wa atubu ilayka (I bear witness that there is no god except Allah Alone, with no partner or associate, and I bear witness that Muhammad is His slave and Messenger. O Allah, make me one of those who repent and make me one of those who purify themselves. Glory and praise be to You, O Allah, I bear witness that there is no god but You, I seek Your forgiveness and I repent to You). 
For a complete description of wudu, please see this answer. 11497 .
For more about wudu-related issues, please see this category: Ablution before Prayer.
And Allah knows best.