1. al-Mustʻaribah (المستعربة) "Arabised Arabs": The term Arabised-Arabs can be used in three different cases:

• Is used for defining the Arabs who are descendants of Abraham through his son Ishmael through his Son Adnan, and they are known as Adnanite: it is defined to the Arabs who settled in Mecca when Abraham took his Egyptian wife Hagar or (Hajar) and his son Ishmael to Mecca. Ishmael was raised up with his mother Hagar and one noble Arab family who left from Yemen and settled in Mecca after the rainless took part in Yemen at that time). Ishmael learned Arabic language and he spoke it fluently during his life. And that is the main reason for calling this Arab group as Arabised. It is believed also that the Prophet of Islam Mohammad is descended of Arabised-Arabs. Some famous noble Adnanite Arab families from this group are: Aldosari, Alharbi, Almaleek, Bani khaled, Bani kolab, etc.

• The term Arabised-Arabs is also used for defining the Arabs who spoke other Afro-Asiatic languages . They are Arabic speakers and regarded as Arabs in contemporary times.
• Bedouin-Arabs whom came from the Assyrian desert are also known arabised also.

Reference: http://en.wikipedia.org/wiki/Arabs