What is the role in the government or social of this Queen in UK in these days?
This site sums it up pretty well:
"As Head of State, The Monarch undertakes constitutional and representational duties which have developed over one thousand years of history. In addition to these State duties, The Monarch has a less formal role as 'Head of Nation'. The Sovereign acts as a focus for national identity, unity and pride; gives a sense of stability and continuity; officially recognises success and excellence; and supports the ideal of voluntary service."
Essentially the queen of England is a figure head without much political or executive power. She has more of a leadership role than an administrative role. She is a voice of influence and guidance more than a voice of power. She also performs many of the ceremonial duties that the US president performs like giving medals of honor and such.