A History of the Arab Peoples is a great general introduction to the history of the Arab countries and the history of Islam. It was written by the respected British scholar Albert Hourani (of Lebanese Christian descent). While not written by a Muslim, the book’s treatment of Muslims is balanced and fair-minded.
This book does not cover the history of other important parts of the Islamic world, such as Central Asia and the countries of the Indian ocean, therefore it should not be considered a complete history of Islam.