To be more accurate, it is better to state that there was two Acts of Supremacy, one by Henry VIII and the other by Elizabeth I. Here is a brief outline of events that caused the two Acts of Supremacy.
Under the reign of King Henry VIII, there was a major point of conflict between King Henry and Rome. Henry wanted a divorce from his wife, Catherine of Aragon, because he fell in love with Anne Boleyn. However, the pope would not allow this divorce, partially due to theological reason and partially due to political ones. Catherine had connections to the Holy Roman Emperor. In short, she had powerful ties. Due to this fact, King Henry passed the first Act of Supremacy (1534), which declared that the monarch of England was supreme over the English church. In other words, the supreme head of the church was Henry VIII.
This act was repealed under Queen Mary, but reissued when Elizabeth took office in 1559. In fact, Mary's actions were deemed treasonous. At the coronation of Elizabeth, she was declared the Supreme Governor of the church.