Though the Divine Right of Kings has eroded to the extent that British monarchs now have largely ceremonial roles, Queen Elizabeth still has a few duties connected to her nation's government. Officially speaking, Parliament can't open or close without her say-so. As the UK Parliament states, the ceremony of opening Parliament actually begins at Buckingham Palace, where the queen begins her procession to the Parliament building, where she enters via, of course, the Sovereign's Entrance. She enters the chamber of the House of Lords wearing the State Crown and the Robe of State.
The House of Commons is invited to the occasion, though they ceremonially deny any interest three times before filing in the House of Lords. The idea, stemming from the English Civil War, is that the Commons is independent of the monarch, though modern-day parliamentarians are more staid and generally more polite than in the time of the bloody civil war.
The queen is also supposed to close out each session of Parliament. In somewhat more extraordinary times, she can also prorogue Parliament, or put it into a recess where members can't vote, though typically a prime minister has to ask her to do it first. Her power to do so is pretty limited, however, and was made even more so by the 2011 Fixed-term Parliaments Act.
