Australia does not have a president so they do not have President's day. The head of the federal government is the Prime Minister but this person is not directly voted by the Australian people (that is a topic for another post). Rather than having a "Prime Minister's Day", Australians celebrate the birthday of the head of state, Queen Elizabeth II. This is observed on the second Monday in June. There are no special parades or even Queen's Birthday car sales that I remember. It's just a low key long weekend where you get to spend some time with family or get away for a couple of days.
As you can see these two holidays are somewhat similar but unique in their own way. There are several other holidays like July 4th and Thanksgiving that are not celebrated in Australia at all. Then there are others like Easter and Christmas that are celebrated on the same day in both countries. To make things even more interesting there is Father's Day which is celebrated in June in America but not until September in Australia (which means if I want to send my dad a father's day card I have to buy it in June and not lose it before I send it sometime in September. As my Dad will tell you I have not been very successful at this so far.) Thankfully Mother's Day is the same in both countries or else I would be eternally confused.
My hope is to make this Living in America posting a weekly occurrence until I run out of ideas (which should take a while). If you have any ideas or feedback I would love to hear it. On the subject of holidays, which is your favourite holiday?