When public opinion reaches a critical mass, it stops being merely a nuisance for the government and then becomes a political liability or opportunity.  That is, in numbers, public opinion impacts policy because everyone in government would like to keep their jobs for the long term, and you can't do that by repeatedly ignoring public opinion.

In the final analysis, the public's opinion has a large role on the policy making process.  Leaders have to walk a very tight balance between hearing the voices of the people and being enslaved to them.  This is not denigrating either side, but rather recognizing the large role that public opinion plays in the making of policy.  At the same time, leaders have to be extremely vigilant in how they actions will be seen by the people. For example, the recent Health Care debate and how leaders voted on it has been of critical importance in the recent midterm election.  Leaders have had to be extra delicate in explaining their stances on the issue in order to ensure that voers understand and embrace said position.  The opinion of the public impacts how leaders support and initiate legislation as they are mindful of how the perception of each bill proposed and supported could be the leader's last.

Public opinion has a huge impact on policy in a number of different ways. For one thing, public opinion is directly related to coting. Therefore politicians who serve in elected positions must attend to the desires of their constituents if they want to keep win reelection. Also, campaign funding comes from private sources, and candidates know that they must keep those sources happy if they want to secure campaign contributions. Political Action Committees (PACs) are organizations that raise money to elect specific candidates. These groups can be very influential in policy decisions. In Florida, for example, the teacher's union PAC played a huge role in influencing school vouchers. This is only one aspect of the question, however. On a larger scale, interest groups form to directly work to raise awareness and affect policy-making decisions Special interest groups exist who are actively involved in everything from environmental issues to social issues, all having an impact on policy. Finally, in some states, the citizenry has an even greater voice in policy. California, for instance, allows for a large amount of what is referred to as "direct democracy" - in the form of referendums, initiatives and even recall (which the citizens used to remove Gray Davis as governor and replace him with Arnold Schwarzenegger.

There are many ways in which public opinion impacts policy. Strength in numbers and fiscal strength lead the way when it comes to swaying policy.

This usually depends upon the policy area in question.  Basically, policy on an issue that is simple and easily understood and that is emotionally salient to a number of people will be heavily impacted by public opinion.

A perfect example of this is abortion.  There is no way that a backroom deal could be made on abortion.  If some rich person wanted to try to cut some deal with a Congressperson, it would never work.  People understand the issue and care.

But when it comes to more complicated, less salient issues, there is less of an impact.  That is partly because there is less true public opinion on the subject.  Your typical person does not know very much about the relative merits of a cap and trade system as opposed to a carbon tax for cutting carbon dioxide emissions.  They do not know much or care much about how, exactly, banks are to be regulated.

In areas such as this, public opinion has little impact.

Beyond this, you would need to think about the difference between the impact of public opinion on Congress's decision making and its impact on bureaucratic rule-making...  This question is so huge... while in graduate school, I spent a quarter studying the public policy process, and another studying public opinion, etc.  If you can ask a more specific question, I can probably give you a more focused answer.