Today, May 5th, is the anniversary of the Treaty of London, which founded the Council of Europe in 1949. Though the Council of Europe isn't an EU institution, it contributed to the spread of human rights and democratic norms across the continent through the European Convention on Human Rights and the court it set up to aid enforcement in Strasbourg.
So May 5th is a Europe Day, but it's not the only one - there's also May 9th, the anniversary of the Schuman Declaration, proposing the European Coal and Steel Community (it no longer exists, but it was the starting point for what's now the European Union).
Usually these days largely go by without being commemorated on a large scale. This year, some Bloggingportal.eu editors have set up My Europe Week, to launch a blog carnival. You can write, video or photograph to express what you think about Europe Day, and what you think Europe is - or should be - about. Of course, bloggers can do this on their own sites, but non-Bloggers can get involved too (by submitting entries via this form). It doesn't have to be long, or in a certain format or strict theme (though being about Europe helps!).
Update: As Grahnlaw has reminded me, you can also follow and contribute on Twitter with the hashtags #MyEurope and #EuropeDay.