That depends on the country and time period.
In most of the history, the responsibilities fo a queen or princess would be to represent and add glory to the country with their beauty and virtues and to serve to extend the country's power or military support from other countries through their marriage. (If you marry a princess you can expect that her father won't attack you).
In more recent history, queens or princesses often engage in social and charitable work. Additionally, some of them have actual positions (queens regnant), and can be head of state, such as Queen Elisabeth in modern UK.
