First of all, welcome to Rome, to the Eternal City where the history of churches, monuments and other tourist sites make it a timeless destination.
Now I would like to clarify some information where many of you get confused and it is very important to understand these differences:
This term is used to make it clear that we are talking about the Rome airport but in reality the airport is located in the city of Fiumicino.
Rome Airport Fiumicino
This term is very similar to the first one mentioned above but is actually more correct, because the Rome airport is located in the city of Fiumicino.
FCO (Fiumicino Airport)
This term is the abbreviation found on the plane ticket to travel to Rome where you will also find the flight number and this is also correct.
Leonardo Da Vinci Airport
Leonardo da Vinci Airport
This is the name of Fiumicino Airport. We always talk about the same airport and this is also correct.
I hope I was clear, there are many ways in which Fiumicino airport is called. But in reality we always talk about the most important airport in Europe, the Rome airport.