Imagine you are driving and come upon an intersection with an accident. In the middle of the intersection is a police officer who is directing traffic. As you pull up to the officer, you have your right hand turn signal on. The officer shakes his head "no" and points you to the left. Compliant, you turn your vehicle in a direction you don't want to go.
Why? Because the police officer has authority. He may also have power in the form of a gun, but it wasn't his power that made you comply. It was his authority.
The writer of the Gospel of Matthew tells an interesting story about authority in Matthew 8.
A Roman officer comes to Jesus asking him to heal his servant. When Jesus offers to go and heal the servant, the officer says, "Say the word, and my servant will be healed. For I too am a man under authority, with soldiers under me. And I say to one, ‘Go,’ and he goes, and to another, ‘Come,’ and he comes, and to my servant, ‘Do this,’ and he does it.”
The Roman officer didn't ask Jesus to lay hands on his servant, to pray for his servant, or even to ask God to heal his servant. The officer simply asked that Jesus "say the word". Like the police officer in the intersection, Jesus just needed to exercise his authority for the servant to be healed.
I'll be honest, I'm not always comfortable praying with authority. I don't know if it's my own lack of faith or not wanting to presume upon God. However, I am reminded of the Roman officer's statement, "I too am a man under authority". He didn't say, "I'm a man with authority", but rather, "under authority".
Imagine that the Roman officer ordered his soldiers to engage in a battle the officer's superiors hadn't told him to perform. The Roman officer would be operating outside of his authority, not under it. He may have had the strongest, most powerful [power] group of soldiers, but he wouldn't have authority.
From this incredible story, the writer of Matthew continues for the next few chapters showing example after example of Jesus exercising his authority. I am convinced that Matthew is pointing out how Jesus modeled working under authority to his disciples. Why? Because, in Matthew 10, Jesus sends out his 12 disciples saying "heal the sick, raise the dead, cleanse lepers cast out demons"(Matt 10:8). The disciples having seen Jesus operate under authority, are now sent by Jesus to do the very things Jesus had modeled for them. The disciples are now sent, under Jesus' authority, to do the same things.
So, how can I pray with authority? Jesus said he did only what he saw the Father doing (John 5:19). Jesus didn't operate outside of the Father's authority. Jesus spent time listening to, and learning to recognize his Father's voice. He heard his commander's orders and was able to act under the authority he had received.
When we pray, the first question we need ask ourselves is "Do I have a clear direction from the Father?". Only by spending time with the Father, learning to recognize his voice, can we hear God's orders to us and pray with authority.