In the Biblical sense, to be fallen is to have rejected God. We reject God many ways, some more evident than others. The best-known case is the original sin of Adam and Eve. Instead of listening to their Creator, they rejected Him by their disobedience.
Our disobedience causes us to be in a fallen state. When we do not go to Mass on Sunday, we reject God by putting other things or people before Him. By disobeying any of God’s commands, we reject God and put ourselves further into the fallen state. Whether it be committing adultery or looking at a woman lustfully, these both are works of disobedience, though in different degrees.
Now, our fallen state comes from the original stain of sin that we inherited by our first parents. God’s grace is what pulls us out of our fallen state. First, we receive baptism which erases the stain of original sin, but does not remove our tendency toward sin. Like bleaching a white shirt, we remove the stain, but we are still likely to stain it again. This is why we have the need for confession. There, Jesus removes the stains of sin that we add to ourselves.
We may be fallen and have rejected God in what appears to us in the most minor of ways, but God wants to cleanse us of our fallen nature and give us a clean soul.