The Labour party was elected to do what the people want intheory.
Or is it just they decide what they want to do, put it in a manifesto, and the public vote then agree to it.
Having said all that, I assume a lot of the public would like the men and women fighting in Afghanistan to come home safely so why can those who want that not petition the Government to do so? And they have to do it?
At the general election, all parties lay out their manifestoes, saying what they would do with the next 4/5 years.
Manifesto commitments are not binding - although they take a lot of flack if they don't keep to them.
The theory is that people then vote for the party which is promising to do what they would want them to do (although actually people choose which party to vote for for all sorts of odd reasons )
But - on your petition point - it doesn't then follow that if the majority of a country want / don't want something, then it must happen. Once a parliament is elected, it makes decisions until the next general election. Even if it could be proved the majority of people wanted something - unless the government actually asked the question itself (by calling a referendum), it's still up to the government.
It is generally accepted to be the case, for example, that a very large number of people would like to see capital punishment return. But no major political party does, so it's not an area of political debate.