Distance-wise, Dublin Airport to the city centre is about 12km (7.5 miles) and the trip will take anywhere from 30 minutes up to an hour, depending on the time of day and traffic. You'll note the word 'traffic', as bus, car and taxi are currently the only modes of transport to and from the airport into the city.
The easiest option would be to take a taxi, which can cost approximately €25-30, depending on time of day and traffic, and other various factors. You can check these out at
Taxifarefinder.
Another option would be to take the
Dublin Express, which services directly to Dublin's most popular locations. Furthermore, it also offers a bus to Belfast from Dublin Airport.
A final option is the
Aircoach (or as locals call it, The Blue Bus), which services the city centre, as well as other areas around Dublin; useful, if you're staying somewhere outside of Dublin's central area. This service also
costs €6 and there's no hassle with having to manage public transportation with your luggage. It runs every
15-30 minutes with three main stops in the city centre: Drumcondra, O'Connell Street and Trinity College.
Hope this helps, and enjoy your break in Dublin!
(Prices quoted are correct at time of publishing)