In Jesus' Name

What does Jesus mean when he says John 14:13-14 "Whatever you ask in my name, this I will do, that the Father may be glorified in the Son. If you ask me anything in my name, I will do it."  And does saying "in Jesus name" qualify?

Acts 2:38 (ESV)  

    And Peter said to them, "Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit.  

Acts 3:6 (ESV)  

    But Peter said, "I have no silver and gold, but what I do have I give to you. In the name of Jesus Christ of Nazareth, rise up and walk!"  

Acts 9:27 (ESV)  

    But Barnabas took him and brought him to the apostles and declared to them how on the road he had seen the Lord, who spoke to him, and how at Damascus he had preached boldly in the name of Jesus.  

Acts 16:18 (ESV)  

    And this she kept doing for many days. Paul, having become greatly annoyed, turned and said to the spirit, "I command you in the name of Jesus Christ to come out of her." And it came out that very hour. 

It seems to me that in the name of Jesus is similar to a policeman saying "in the name of the law" and it means by the authority of Jesus. What do we have the right to ask God to do by the authority of Jesus? Are there any examples of the apostles asking God to do something by the authority of Jesus?