That's exactly what we have now - a Monarch that is a leader symbolising unity, traditions and heritage, while the actual governing (ruling) is done by the Government.
The Queen is a constitutional Monarch and her powers are severely limited; he can't really do much without the Government's consent.
Britain hasn't been an absolute Monarchy for many centuries and was, in fact, one of the earlier European democracies in modern times; among others, it is the birthplace of Parliament.
It's system of Government is very much like that of Sweden, Norway, Spain, the Netherlands, Belgium, Luxembourg, Denmark and other European Constitutional Monarchies.
The British Monarch holds a constitutional position of Head of State. According to convention, the Queen's powers are exercised upon the advice of her Privy Council (which includes the Prime Minister). In practice, political power is exercised today through Parliament, the Prime Minister and the Cabinet. The Monarch holds a weekly audience with the Prime Minister, as well as regular audiences with other members of the Cabinet.
In theory, the Queen has vast powers, however most of them haven't been used for decades, if not centuries. The following is a partial list of Queen Elizabeth’s Royal Prerogative Powers:
- The Queen alone, as Head of the Armed Forces, may declare war or peace
- The Queen alone may conclude treaties
- The Queen (as commander-in-chief) may choose and appoint officers of all ranks
- The Queen may convoke, adjourn, remove, and dissolve Parliament
- The Queen may appoint a Prime Minister of her own choosing
- The Queen may dismiss the Prime Minister and his Government
- The Queen can choose and appoint all judges, councillors, officers of state, etc.
- The Queen may initiate criminal proceedings, and she alone can bestow a pardon
- The Queen may refuse the Royal Assent
- The Queen may refuse to dissolve Parliament when requested by the Prime Minister
- The Queen can choose and appoint all Archbishops, Bishops, and ecclesiastical dignitaries
- The Queen may exercise the refusal of the “Queen’s Consent” (direct Monarchical assent is required for a bill affecting the prerogative, hereditary revenues or the personal property or interests of the Crown to be heard in Parliament).
- Since the Sovereign is “first in honour, dignity and in power--and the seat and fountain of all three,'' the Queen may bestow all public honours, including creating peerages or bestowing Orders of Chivalry
Arguably, her non-political, ceremonial roles are far more important than the political ones since they are the REAL ones. Those roles include:
- Perform the ceremonial and official duties of Head of State
- Represent Britain to the rest of the world
- Provide a focus for national identity and unity
- Provide stability and continuity in times of change
- Recognise achievement and excellence (by means of awards, medals or orders)
- Encourage public and voluntary service
- Support charities and foundations and highlight their causes
EDIT: The Prime Minister is not elected by the House of Commons.
The leader of the political party that gets most votes (conservatives, labour, liberals) is invited by the Monarch to form the new Government (become the Prime Minister). Theoretically, the Monarch may chose someone other than the leader of the party with most votes (appointing the Prime Minister is one of the Monarch's prerogatives), however that hasn't happened for centuries.
The House of Lords consists of 3 types of representatives:
1. The Lords Spiritual - 26 senior bishops of the Church of England.
2. Life Peers - they are appointed by the Monarch on the advice of the Prime Minister. The Monarch's role in this appointment process is rather symbolical.
3. Hereditary Peers - not all peers have the right to sit in the House of Lords; currently, of the 789 representatives of the House of Lords, only 90 are hereditary peers.