The eNotes/Wikipedia page on the subject defines fine art photography in many ways, but I think the best definition is quoted on this page by Bernard Jones from Cassell's Cyclopedia of Photography. He says that art photography is any photograph created by an artist with the intention of doing something more than merely creating a realistic rendering of the subject. He says the piece has to attempt to convey a personal impression.
I would also say that a photograph could be considered fine art photography if the artist’s intention is to do something other than create a piece of photo journalism or a piece of advertising photography. The ultimate goal of photo journalism is to portray accurately to the public exactly what is happening at a given time. The ultimate goal of advertising photography is to create a photographic image that will meet the needs of an advertising campaign where the primary goal is to sell a product.
A piece of fine art photography must also be held to the same standards of quality to which every other piece of fine art is held. The elements and principles of art must be used well to achieve a pleasing image.
Based off of the article I read by A. Cemal Ekin, they highlighted certain points of what they believe fine art photography is. The first point was message. Fine art photography should show some kind of message, commentary, just something to give the photo a bit of story even if it's a mystery photo leaving you with many questions. The next point that they showed was the intention of the photograph. This is essentially knowing what you are trying to make a photo like; being prepared. This is important to keep out the hundreds of random photos that need to be deleted if pictures were not thought out well. Then there is choice. What are you choosing to have the photograph look like? What are you settings? What are you trying to photograph? These types of choices. Lastly, there is technique. How was the photo edited?
Fine art photography is a mixture of all these different things. The author of the article included print as one of their points, however living in such a technological time allows something to be considered fine art without it needing to be printed in some form or another.