There is no god except Allah and Muhammad is a messenger of Allah
This is the statement that qualifies someone as a Muslim. If you are convinced of this at some point then you are a Muslim.
Formally someone becomes a Muslim by saying - in front of witnesses - "I witness that there is no god except Allah and that Muhammad is a messenger of Allah."
To understand this statement fully some explanation is needed of the word 'god'. A god is understood to mean anything or anyone who you use to set your values in life; to determine how you judge a matter as good or bad and how therefore you live your life. A god is also someone or something that people try to keep happy so that it may be able to change their lives and make those people happy. Since every human being by nature forms value judgements everyone has at least one god.
What is your god? Is it your desires? - Is it your work? - Is it fashion? - Is it your family? - Is it your nation? - Is it your church?
Whatever your gods may be, becoming a Muslim frees you from the service of them and brings you to the service of Allah and only Allah.
Believing that Muhammad is the messenger of Allah flows from believing in the Qur'an.