All 16th June, Dublin remembers this same date in 1904 in Leopold Bloom's life, the main character of James Joyce's novel Ulysses.
This day is known as Bloomsday. The novel is the excuse for Dubliners and their visitors to relive Bloom's adventure which takes place during a day in different settings in Dublin. Thus, it's a tradition to visit his birthplace, have a walk along the Liffey or go down to the beach of Sandycove. If you go to the city centre, don't miss O'Connell Street, Belvedere school or Monto. These are remarkable landmarks in Joyce's life and novel and the scenery for many performances as well.
If you want to enjoy, a look on Ulysses is a great idea to start with.
More information, Bloomsday Festival 2010