The storming of the Bastille was a major symbolic event in the French Revolution. It showed that the people of Paris were willing to come out and support the revolution against the royalists who were trying to stop it. The Parisian crowds had weapons that they had gotten from a government armory, but they lacked powder. The powder was in the Bastille, an old prison in Paris. The crowd went there to demand to be given gunpowder. The government refused and the crowd stormed the prison, eventually defeating the defenders. This was a major showing of the people's determination to protect the revolution and the National Assembly that was bringing it about.
It happened on the 14th of July 1789.
The third republic, almost one century later, decided to have the national fest on this date.