The history of petitioning the government goes back to Imperial China. It continued with petitions to the House of Commons in England. Probably the best-known petition of all time is the Declaration of Independence -- a petition to then King George to free the colonies. The Declaration of Independence, however, is not law. Legally, the right to petition is perhaps most well-known for being enshrined in our own constitution's First Amendment, which:
guarantees the right of the people "to petition the Government for a redress of grievances." The right to petition has been held to include the right to file lawsuits against the government.
Note that the common phrase "You can't sue the Government" is flat-out wrong. You can and it's guaranteed in the constitution.
One such lawsuit, Johnson v. Department of the Treasury of the United States, et al., case No. CV11 6684 (NJV), "alleges suppression of debate re great benefits that the government would automatically gain from replacing Federal Reserve notes with new issues of United States notes."
You can read more about it here: http://www.opednews.com/articles/The-American-Crisis--To-F-by-Clifford-Johnson-120103-997.html
In the modern age, it is unusual for a group of people to march up the steps of City Hall, or the Capitol, with thousands of signatures on a petition, and attempt to hand-deliver it to their representative -- though if you can get past the security (blocking you from your constitutional right to redress) and long lines, you will probably get noticed!
But today, mostly petitions are done on-line. If done with the proper organization, and well-publicized, these can be effective. Unfortunately, the optimum process keeps changing and what worked before does not work as well today.
For example, I have a long-standing petition to reintroduce United States Notes -- aka Lincoln Greenbacks -- as a sovereign debt-free currency -- on Change.org, one of the older, and previously, most popular, e-petition sites. Recently, the number of signatures passed 500, thanks in part to a recent surge in Twitter support of retweets from a Twitter account I set up - @NewThinking2 - about 6 months ago, which is only now beginning to gather strong support. It takes a while to build support on the Social networks, and it's better to do this before launching a petition, particularly if it's going to be on one of the sites that has a time deadline for gathering signatures, like Whitehouse.gov, which requires 100,000 signatures in 30 days in order to get an Administration response. This can be a very high bar to hurdle, particularly if there are other e-petitions similar to yours out there already; on the other hand, the Administration may respond to a set of similar petitions all at once.
The shifting landscape of successful e-petitioning has another problem. What happens if you've already gathered a lot of signatures and another, better, platform, comes along, or in the real-world case of Change.org, which dropped features like automatically addressing all members of Congress and the Senate from its petition options? Then, you are left with either using a hobbled product and keeping the base of signatories you already have, or having to start all over again, perhaps contacting all the people who signed your original petition, to sign the new version somewhere else.