Read: Luke 4:31-37
...and they were astonished at his teaching, for his word possessed authority. (Luke 4:32 ESV)
We all know and recognize authority. Even if we don't like that authority or feel like it is being abused we know authority when we see it. Official representatives of our government carry authority. Officers of the law, firemen, first responders, and other emergency personnel carry authority. Even the IRS has authority.
Authority is power. It is a voice of order. When authority speaks, those of us within the sound of the message have to choose how that we will respond. Compliance or rebellion? The trick I think sometimes comes in recognizing and validating authority. It's fairly easy if a policeman pulls up behind you with their lights flashing and siren going to recognize the authority they should have over you. Sometimes in the world of Christian faith this seems to get a little unclear for people.
As Christians our ultimate authority is Jesus. The demons he cast out recognized his absolute authority. As such, we should filter life's big decisions, personal victories, and private troubles with Jesus. He already knows them, but his authority works in our favor. In fact, in some ways it actually extends to us. Jesus cast out demons because of his authority, and for nearly two millennia his followers have done the same thing while operating as heavenly representatives of his authority.
Also, chances are pretty good that God has placed us under the authority of a fellow believer. Timothy was under Paul, even as Paul had submitted himself to the authority of the disciples and other Christian leaders such as Peter and James. Rare indeed is the occasion that God places us in a position to love and serve Him without living under authority. It's unhealthy and dangerous. Who is your authority? How are you submitting to them?