I am familiar with three kinds.
Popular petitions are for political purposes.
They carry no legal weight, but can be a strong statement to elected officials that a lot of people support or oppose a proposed development. Legal Protest Petitions are applicable in only some circumstances, but when they are applicable, they are a powerful tool. For instance, in Overland Park, KS, submission of a valid Protest Petition forces the developer to get 9 out of 10 City Council votes, instead of a simple majority. In Overland Park only rezonings, special use permits, or plans which so differ from the original that they are treated like a rezoning may have protest petitions. In the fight against Wal-Mart at 159th and Metcalf in Overland Park, KS, we used both types.