How is public opinion relevant to a political system?
There are actually two perfectly correct answers to this question, the first being that public opinion has no relevance whatsoever, and the second being that it has great relevance. Let's begin with a philosophical discussion about the nature of political representation, followed with a discussion of the practicalities involved in running for office, and then move on to an explanation of both answers.
Do people elect their representatives to make decisions the way they would? Or do they elect their representatives to exercise their own judgement on matters? I think there was a time when most people elected representatives and trusted them to exercise their judgement on behalf of the people, but that this expectation is less the case than ever before. When our country began, it was far more homogenous than it is today, mostly rural, mostly farmers, and elected officials were generally more educated and wealthy than those who elected them. There was no media such as there is today, and people were not necessarily all that well-informed on all the issues. So as a general matter, those elected were expected to exercise their "superior" judgement for the good of all. Today, there seems to be a tendency when one votes to expect the person elected to reflect the voter's opinion, rather than exercising his or her own judgement. This is a thorny problem, an ethical dilemma, really, for a conscientious official, no matter what the expectations of the populace are, and the best politicians must grapple with this dilemma almost daily. I am not aware of any magic formula that can aid an official in deciding which to do.
As a practical matter, today politicians cannot hope to get elected at all, to represent the will of the people or to exercise their own judgement, unless they make representations that reflect the will, at least to some degree, of the people who vote. This is being borne out right now in the United States, as candidates posture to obtain nominations in the primaries to run for the presidency. If they do not promise to render policy as their constituencies desire, they cannot hope to run in the general election. Once they run in the general election, they must appeal to a larger constituency in order to win, and the contradictions between their promises in the primaries and in the general election are quite extraordinary.
Having said all that, I have to say that since we are such a diverse society now, with constituencies on social, economic, domestic, and foreign policies, in addition to all of the special interest groups in our society with conflicting agendas, on some level, public opinions can tend to cancel one another out. We are so fragmented that there are simply too many opinions that contradict one another. An elected official who is trying to decide what to do about an issue could very well find his or her constituency at a 50/50 split, or even a further split amongst several different options. And with all of our blogs and tweets and Facebook postings, sometimes I think we are making so much noise politicians should simply ignore us.
However, politicians do not ignore public opinion, and polling is a fact in today's political landscape. Elections aside, in which polls are paramount, politicians are frequently taking the pulse of their constituencies and can find that they disregard them at their peril. Those who are most zealous about their representatives keep tabs on their voting records, for example, and those who do not vote to reflect the views of the zealots are threatened with a defeat in the next election. When there is a clear majority opinion on a matter, or even when there is a vociferous minority, public opinion does matter. One recent example is the Supreme Court ruling on same-sex marriage (and do not believe for one moment that the Supreme Court is not a political institution.) A majority of Americans favor same-sex marriage, and the Supreme Court was completely aware of public opinion when it made its determination.
So, the answer to the question really depends upon the political context, whether or not there is some clear majority opinion, and whether or not a politician believes that he or she must represent the will of the people or should use his or her own skills, experiences, and education to make policy determinations. Sadly, very few politicians today have the courage to do the latter, so all of the opinions that we so loudly broadcast do matter, for better or worse, when they don't cancel one another out.