Dublin Airport is located to the north of the city centre near the M50 and M1 motorways. You've got three options when it comes to getting to and from the airport: bus, taxi or car.

Buses:
Dublin Bus run a handy airlink express service from Busaras, the headquarters of the bus network and O'Connell Street to and from the airport. The buses are clearly marked 'Airlink' and are numbered either 747/8. (after the Boeing jet, I presume.) 747 Airlink buses depart the airport every 15 minutes from 05:45 until 07:30 and then every 10 minutes until 23:30 Mon-Sat. On Sundays, first buses depart at 07:15 and run every twenty minutes from then on. The 748s depart every 30 minutes. The fare is €5 for an adult and €2 for a child which can be paid to the driver. The bus stops are located right outside the arrivals main exit.

The 16A, 41, 41B, 41C, 46X, 58X & the 746 all take you from the airport to the city centre and the times are printed on the bus shelters.

'Aircoach' also run a service that runs 24 hours a day that brings you from the airport to the city centre and the south of the city stopping at the major hotels & various street points. The blue coaches of this service are easy to spot as they, as opposed to the bright multi-coloured double-deckers of Dublin Bus. Likewise, you'll find an 'Aircoach' bus-stop right outside the arrivals exit. The fare is €5 and is paid to the driver.

Click here for more info on Aircoach.

'Aerdart' run a shuttle bus service from the airport to Howth Junction DART (Dublin Area Rapid Transit) station and back. You can then travel from here on the light rail line into the city centre. The DART line also services the south (and north of the city) Buses depart every 15 minutes and the fare is €5 for adults and €3.15 for kids and you can get off at whichever DART station suits you best. 
     
Taxi:
There are always taxis at the rank again just outside the main exit for arrivals. Unlike in some countries, this is a generally safe way to travel, but you should ask the driver to estimate the cost of the fare to your destination first. The average cost of a fare to the city centre is €15. There are, of course, additional charges for extra passengers, luggage, animals, time of day, Sunday and public holidays.

Car:
If you want to rent a car you'll find Avis, Hertz, Budget, Murrays Europcar and National Car Rentals desks in the arrivals concourse. There are also pre-booked car-hire companies like Argus Rent A Car and Malone Car Rental who operate desks there too. Opening hours for all are 06:00-23:00 (or the last scheduled flight). ECL operate a chauffeur-driven saloon and luxury minibus service from their desk. The car-hire charges vary dependant on whether you want a small runaround hatchback or a prestige saloon and whether you're insured to drive or not.

Now that you've got your vehicle, the way to the city centre is adequately signposted on your way out of the airport grounds. If you're returning to the airport from town, you can go up to Drumcondra and keep on the main road past it. Continue straight to get onto the M1 ring road and take the left turn off the roundabout to the airport. Don't worry, it's well signposted.