The high school civics explanation is not the whole story. Yes, a law is drafted, and read out from committee. It is then passed by the House and Senate and reconciled between the two. Then the President signs it, in the smoothest case. This outline approximates the mechanics, but does not cover the motivational forces which create law in America's democracy.
Something happens that prompts a need for a new law. A security crisis or a securities crisis, a new technology, or a rash of some kind of crime, leads a group of people to think about changing our laws. We have laws named after victims, like Megan's Law, and laws named after legislators who saw a need, like Sarbanes-Oxley.
We have minor laws which benefit small groups of people or answer temporary needs, and sweeping laws that change the course of American history. Consider the Civil Rights Act of 1964: it changed the world. Because we are a pluralistic society, most laws begin when some interested group begins to push for legislation.
Pluralistic means that we are a nation of competing interests, which contend to shape our continually evolving laws. We are not, and our founders did not intend us to be, one huge shapeless mass which shifts its monolithic opinion at the same moment. On the other hand, we are not indifferent puppets of opinion-molders either. Because we do not have to be afraid of our government, we are free to think about the direction we would like it to take.
Listen to people in bars, at work, or after church. These conversations are made of people sharing and forming opinions. Sometimes the places where people gather and talk are places where new movements begin, where laws begin. Some of the opinions passed around may seem childishly ridiculous or pitifully uninformed. Some though, sometimes, contribute, down the road, to shaping law.
There is more to law-making than talk, of course. Talkers have to organize. They may write to their local legislators with their ideas. They may contribute to a website, or to a political party. Or they may circulate a petition, as Megan's parents did. They may only contribute their voice to a rising murmur of public opinion that will swell into actual law some day.
Laws are made of ideas, whether hopeful, fearful, or of cold calculation. In America, laws are an actual expression of the people's will. It would be wrong to say that every citizen has a duty to contribute to the evolution of the law, but every citizen should be aware of a high moral privilege.