What is "apostolic authority"? Some Christian traditions use the term "apostolic authority". What is meant by this phrase? Does it have more than one meaning depending on which doctrinal tradition it is coming from? Are there any traditions that don't hold it has a meaning at all?

Do groups that hold such a view think that some things can only be carried out by apostles with specifically granted authority?
In broad terms there are two schools of thought on this. One concerns apostolic succession - the idea that the authority of the original 12 apostles has been passed down through the generations, relay-race style. For those who follow the doctrine of apostolic succession, apostolic authority belongs only to those who have inherited it from a predecessor, which in theory can be traced back to the original twelve.

The second notion of apostolic authority is that anyone can be ordained (I use the word in a very loose sense here) by God to have an apostolic ministry. In it's broadest interpretation it could include all Christians, because we're all called to act the way apostles do - but more often than not it's applied to individuals who demonstrate the characteristics of an apostle in a very clear way that is recognised by their peers and others.

Dr. Martin Schmaltz provides this succinct definition:


  Apostolic Authority is being sent by Jesus Christ with an
  authorization to use His inherent power to fulfill His purpose. This
  apostolic sending has four parts:
  
  
  First, the source of authority comes from outside the one operating in authority.
  Second, authority has a specific area or place to operate.
  Third, there is usually a specific mission or purpose for the use of authority.
  Finally, it requires obedience and complete follow-through for authority to be effective
  
  
  Apostolic Authority is acting as Jesus' representative to this world
  to accomplish His purpose.


while Dr Steven Lambert offers a much more lengthy answer to the question, which also deals with people who self-identify as having apostolic authority. He points out that different churches will recognise different people as having apostolic authority.