Act Mode is where the Master directly fights the monsters and fiends of Tanzra. To do so, the Master sends his essence into a statue of himself, which becomes his avatar. With this body, the Master sets out to rid the land of evil. Though the Master himself is eternal, his corporeal form is not.
There are two Acts in every area, each one culminating in a boss fight. One Act occurs right away, to make the land habitable. Once it's been cleared, a temple will be erected to serve as place of worship for the people. The second Act occurs after a significant event has happened in the story, usually accomplished by purging Monster Lairs first.
Note that the Score doesn't reset to the last checkpoint if the Master falls in battle. With careful application of Sources of Life, and perhaps a 1up loop, the Master can drive his score up really high.