Okay in most democratic countries the role of Head of State and Head of Government is split up. The USA is a bit of an oddball it puts them together. The Head of Government (normally a Prime Minister or Chancellor) wields political power while the Head of State (wields constitutional power and sometimes some political power). The Head of State is normally either a President or a Monarch.
Today the Monarchs role is limited to Constitutional, Ceremonial and Ambassadorial.
Constitutionally the Monarch has the right to be kept informed, advise and warn the government of the day. As a position they hold for life they can then pull on a life in the role to execute these duties. The Queen does this through regular meetings with government ministers, state papers and a weekly meeting with the Prime Minister. The Queen also retains some personal Royal Prerogatives these are to call and dissolve parliament, appoint the Prime Minister and give Royal Assent to bills so they can become law.
Ceremonially the Queen also has to rubber-stamp the use of other prerogatives like declaring war. She also acts as the amplification of the nation in a living breathing person and fulfils traditions and ceremonies like the state opening of parliament.
Ambassadorial the Queen greets foreign delegates and goes on regular overseas trips to maintain and strengthen good relations with other nations. She and the Royal family also champion many charities in the name of the nation.
The Monarchies personal wealth all comes from land. The Crown estates produce £220 million a year however this is all surrendered to the Government in exchange the government takes on all the expenses of the Monarch in their role as Head of State. Personal incomes comes from the Duchy of Cornwall (which goes to the Prince of Wales or to the Monarch if there is not one) and the Duchy of Lancaster (which goes to the Monarch). The Monarchy also has a variety of other estates and investments which are private.
The British Monarchy has evolved over time. Previously in England but also in the other constituent realms the Monarchs were forced by the nobility and the church to have parliaments which started off with just making sure the King did not act too tyrannical overtime it extended to represent the commoners and slowly weakened the power of the Monarchy. The English Civil war and the Glorious Revolution (not a violent one we are too British for that) were major transfer of powers to parliament. If you want to know more about them (http://en.wikipedia.org/wiki/Glorious_Re…