Dublin has two Cathedrals: one for the Catholics and one for the Protestants. Both are worth a visit and give you a real insight into the different attitudes amongst the two branches of Christianity.
Christ Church Cathedral is located up near Dublin Hill and is a fine example of 12th century Cathedral building. The grounds are small but provide a break from the surrounding roads. The Cathedral itself is impressive, and you're able to wander around it. The Protestants like to remember great war dead, so you'll find many statues and plaques.
Housed across the road from the Cathedral, and with an interesting bridge over the road, is Dublinia. This is an exhibition of Viking and Norman artefacts and is worth visiting whilst there.
Not too far south of Christ Church is St Patrick's Cathedral, founded around the same time. The Catholic Cathedral has far more relics, statues, plaques etc... You'll also see that it is more grandly decorated. Again it is a fine example of Norman architecture. Interestingly, the Catholics like to remember their great authors and scientists, so you'll find much dedicated to the likes of Swift and Boyle.
If you like Cathedrals then it is important to visit both sides of Dublin's faith. Both Cathedrals are fine architecturely and provide an interesting insight into the differences between the two strands of Christianity.
Dublin Castle is situated near Christ Church. Whilst parts date back to the 13th century, most of it was rebuilt in the 19th century. It's not very big and not very castle-like. It is really a stately home that is a quirky mix of architectural styles. It has a 'lego' feel to it. As a building it is not that impressive, but the state apartments and the grounds may be of more interest to some.