A constitution is a country's set of most important rules. It says who runs the country, how leaders are chosen, and what rights every person has - like the right to go to school, to say what you think, and to be treated fairly. Most countries have one, often written down in a single document.
Denmark's constitution was signed on 5 June 1849. Before that, the king of Denmark made most of the decisions himself. The new constitution said that from now on, the country would also have elected politicians (the people choose them) who would help run things. The king stayed - he just didn't run the country alone any more.
On 5 June, Danish flags are hung up everywhere. The flag, called the Dannebrog, is one of the oldest national flags in the world - a white cross on a red background, used for over 800 years. Many Danes put a little Dannebrog on the kitchen table at every birthday and special day.
Constitution Day is celebrated with picnics, public speeches and family days out. Many schools mark it just before the summer holidays start. It is also Father's Day in Denmark - a happy coincidence, so lots of Danish dads get cake and breakfast in bed on the same morning.