A fantail balcony is a balcony cabin on the fantail (back end of the ship). Not all cruise ships have these, but RCCL have ships that do.
The advantage is like p40whk said... you get a better view because you can see more without the sides of the ship interferring. Is it a make or break item? Probably nice, but I think her point was that if you go with a knowledgable cruise travel agent, its these little bits of knowledge that can make you cruise better.
Don't worry about missing the shows for a late dinner. They have two sittings for the shows also. Pick a dinner time you are most comfortable with. We always get an early sitting because my wife gets sick to her stomach if she eats late. Thats the beauty with cruising - you can choose.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise, Magic 1/14
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013