Barcelona airport is l'Aeroport del Prat (phone: 902 400 500), in El Prat de Llobregat, about 12 km from Barcelona.
To move between the airport and the city, you have the following options:
The Renfe station is connected to the Airport, and a train to Barcelona runs every 30 minutes (it takes 25 minutes to arrive at the Plaça Catalunya stop). Price is 2,20 € and it stops also at Sants Estació, Plaça Catalunya and Arc de Triomf (you can access the metro from those stations). It runs from 06,13am to 10,45pm (there is a later train but it doesn't stop at every station).
You can check the timetable and current prices here: Renfe timetables
The Aerobús runs every 12 minutes from 6,30am to 12am. It costs 3 € and stops at Plaça Espanya, Plaça Universitat and Plaça Catalunya, all in the city center.
There is also a Nitbus (Night bus) that leaves at 10,15pm, 11,35pm, 0,50am, 2,05am, 3,20am and 4,22am, but it only goes to Plaça Espanya.
Taxis in Barcelona have very characteristic black and yellow colors, so it's not hard to see the taxi ranks outside the terminal. A trip from the airport to the city center costs around 24 €.
The main car hire agencies have booking points at the airport.
Barcelona Airport to Barcelona City Hotel Private Transfer
If you don't want to deal with all the problems of arriving at an unknown airport and finding the bus stop or a taxi, you can book a private transfer for 63€-111€ (depending on the number of passengers and day and hour of arrival).
Book your private transfer here.
Some high-end hotels offer an airport shuttle service to their customers, so don't forget to check this when you book your room.