Lexically, Tawaaf means to turn, walk, etc around something. Islamically, Tawaaf is the circumambulation of the Ka'bah seven times in anticlock wise direction. People usually do this during umrah or hajj. It is the second compulsory act of Umrah. If one omits this act whether knowingly or because of ignorance of the rule, the Umrah becomes null and void, and as a result one's Ihraam also becomes void. A Tawaaf is a complete act of going round the Kaaba seven successive times.
Tawaaf al- Qudoom is the Tawaaf of arrival for Hajj. This is done by the pilgrim who has entered ihram for Hajj and by the pilgrim who is doing Qiraan, i.e., joining ‘Umrah and Hajj, when he reaches the Ka’bah. It is one of the obligatory duties or Sunnahs of Hajj – there is a difference of scholarly opinion. 
(1) Intention (Niyyat)
(2) Formal purification i.e. wudhu is farz(compulsory) according to some scholars and it is Mustahab(recommended) according to other scholars. If you perform wadhu then there is no difference of opinions. 
Except for the specific supplication between the last stretch of each ‘Tawaaf’ of the Kaaba, ie. between ‘rukne Yamani’ and ‘Hajar Aswad’, there are no specific or particular duas recorded in the authentic Sunnah which one needs to recite during the ‘Tawaaf’. One may praise The Lord Most High and/or supplicate the Lord Most Merciful in any language and words in the ‘Tawaaf’. It is Sunnah to recite this supplication during the last stretch of each ‘Tawaaf’ between the ‘rukne Yamani’ and the ‘Hajar Aswad’:
‘RABBANA ATINA FEEDDUNYA HASANATAN WA FIL-AKHIRATI HASANATAN WAQINA AZAABAN NAR.’ Qur’an. Surah Baqrah 2:201
(O our Lord, give us the good in the world, and give us the good in the Hereafter, and save us from the torment of the Hell Fire.) Narrated by Abu Dawood, 1892; classed as hasan by Shaykh al-Albaani in Saheeh Abi Dawood. 
There are many types of Tawaaf around the Ka’bah:
Tawaaf al-ifaadah during Hajj, which is also called Tawaaf al-ziyaarah (Tawaaf of visiting). It takes place after the standing in ‘Arafah, on the day of al-Adha or after it. It is one of the pillars or essential parts of the Hajj.
Tawaaf al-wadaa’ (the farewell Tawaaf), which is done after completing the actions of Hajj and when one has decided to leave Makkah al-Mukarramah. It is obligatory, according to the more correct of the two scholarly opinions, for all pilgrims except women who are menstruating or bleeding following childbirth, and the one who fails to do it must sacrifice an animal of the type that is valid as an udhiyah.
The Tawaaf of ‘Umrah. It is one of the pillars or essential parts of ‘Umrah, which is not valid without it.
Voluntary Tawaaf: Tawaaf in fulfilment of a vow made by one who vowed to circumambulate the Ka’bah. It is obligatory because of the vow.
Each of these Tawaafs must include seven circuits, after which the person offers two rak’ahs behind Maqaam Ibrahim, if possible; if he cannot do that then he may offer the prayer in any other part of the masjid. 
1. On the day of their departure, some pilgrims’ go.to Makkah to perform their Farewell Tawaaf before throwing the pebbles at the Pillars. Then they return to Mina to throw the pebbles, and depart from Mina for their respective countries. Thus their final rite becomes that of throwing the pebbles at the Pillars and not of the Tawaaf of K'abah. This is an error as the Prophet (peace be on him) said, No one should depart without his last visit being to the House (K'abah) ''
Accordingly, the Farewell Tawaaf must take place after one has completed all the rites of hajj (and before starling his journey for home). After this Tawaaf one should not stay in Makkah except during the time it takes to prepare to depart.
2. After finishing the Farewell Tawaaf, some people walk backwards, facing the K'abah, as they exit from the Sacred Mosque, under the impression that this is a veneration of the K'abah. This act is an innovation (bid'a)in the religion and is without any basis.