What does the queen do?

A lot. Are you prepared for a long answer?
http://www.royal/gov.uk/ lists her role,what she does each day.Here are some snippets:
"The British Sovereign can be seen as having two roles: Head of State, and 'Head of the Nation'.

As Head of State, The Queen undertakes constitutional and representational duties which have developed over one thousand years of history.

There are inward duties, with The Queen playing a part in State functions in Britain. Parliament must be opened, Orders in Council have to be approved, Acts of Parliament must be signed, and meetings with the Prime Minister must be held.

There are also outward duties of State, when The Queen represents Britain to the rest of the world. For example, The Queen receives foreign ambassadors and high commissioners, entertains visiting Heads of State, and makes State visits overseas to other countries, in support of diplomatic and economic relations.
The Queen's role is to:

Perform the ceremonial and official duties of Head of State, including
representing 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;

Encourage public and voluntary service."
Her working day:

"The Queen begins her ordinary working day like many other people - with paperwork at her desk.

After scanning the daily British newspapers, The Queen reviews her correspondence. Every day, 200-300 (and sometimes many more) letters from the public arrive. The Queen chooses a selection to read herself and tells members of her staff how she would like them to be answered.

This enables The Queen personally to see a typical cross-section of her daily correspondence. Virtually every letter is answered by staff in her Private Secretary's office or by a lady-in-waiting.

The Queen will then see, separately, two of her Private Secretaries with the daily quota of official papers and documents. This process takes upwards of an hour.

Every day of every year, wherever she is, The Queen receives from government ministers, and from her representatives in the Commonwealth and foreign countries, information in the form of policy papers, Cabinet documents, telegrams, letters and other State papers.

These are sent up to her by the Private Secretaries in the famous 'red boxes'. All of these papers have to be read and, where necessary, approved and signed.

A series of official meetings or 'audiences' will often follow. The Queen will see a number of important people.

These include overseas ambassadors and high commissioners, newly appointed British ambassadors, senior members of the Armed Forces on their appointment and retirement, and English bishops and judges on their appointment.

Each meeting usually lasts 10 to 20 minutes, and usually The Queen and her visitor meet alone.

The Queen may also meet a number of people who have won prizes or awards in a variety of fields such as literature or science, to present them individually with their prize.

If there is an Investiture - a ceremony for the presentation of honors and decorations - it begins at 11.00am and lasts just over an hour. The Queen usually meets around 100 people at each Investiture to present Orders, decorations and medals.

The Queen will often lunch privately. Every few months, she and The Duke of Edinburgh will invite a dozen guests from a wide variety of backgrounds to an informal lunch.

If The Queen is spending the morning on engagements away from her desk and other commitments, she will visit up to three venues before lunch, either alone or jointly with The Duke of Edinburgh.

On a regional visit, The Queen and The Duke of Edinburgh lunch with a wide variety of people in places ranging from town halls to hospitals.