+Jennifer E I've had to explain this to a couple of American friends. (+Meirav M. let me know if I get anything drastically wrong for the UK, I'm speaking from a Canadian perspective).
The systems are pretty much identical between the American, French, Canadian, and UK systems, with a couple of surface things that make that slightly shift the balance of power in each region.
* Congress == Parliament
* Senate == Senate (CAN) == House of Lords (UK)
* President == Governor General (CAN) == Queen/The Crown (UK)
The big difference is that at some point in history in, the UK and Canada, the congress (parliament) elected a single representative to act as their voice: a primary-congressman (prime-minister). They then voted to restrict the president's (the crown's) power since it was too dictatorial; rather than face a coup, the president (the crown) agreed.
Overtime, given the president (the crown) had no real decision making power, power shifted to congress (parliament), and as the primary congressman, the prime-minister became the defacto head of state.
Prime-ministers are not elected by the people,
That is also how France can have both a president and a prime-minister.
I hope that makes things less confusing (not more)
EDIT: I have made edits for accuracy based on comments by +Mark Hanson
here in the UK people elect Members of Parliament to represent them, and normally the leader of the party that gets the largest number of MPs is the one who gets to be Prime Minister.
the Prime Minister is the country's leader - so in terms of power, the PM is the UK equivalent of the US President, but the difference is that in the US the President is elected by the general population, whereas in the UK the PM is not.
yes, The Crown's responsibilities equate more to the powers of the The President.
If the PM is not the de-facto head of state, I would say at least de-jure. At least in Canada, The Crown is nothing more than a rubber stamp machine for whatever Parliament puts across its desk.
Which brings me to +Mike Barton The Governor General (in Canada) has the powers of The Crown which are signficant, including introducing new acts before parliament, and denying assent (veto) to any act from parliament. That they choose not to exercise that power does not mean it does not exist.
That the PM appoints the GG in Canada does increase the power of the PM role relative to Britian, and decrease the power of the GG role relative to the US President.