The I-360 petition is one that U.S. Citizenship and Immigration Services (USCIS) provides for foreign nationals wishing to begin the green card (lawful permanent residence) application process within one of the "special immigrant visa" categories (see below), or as an Amerasian, a widow(er) of a U.S. citizen, or a battered/abused spouse, parent, or child of a U.S. citizen or permanent resident.
Most people filing Form I-360 will do so as a "self-petition," while in a few cases, the sponsoring organization files the petition.
Despite the generic-sounding name, special immigrants are a random selection of visa-eligible people, including:
Each of these categories must meet its own special requirements in terms of eligibility and documentation. See Eligibility for a Green Card as a Special Immigrant for more detail.
See Domestic Violence Against Immigrants: VAWA Protection for the rules on getting a green card as a battered spouse or family member.
First the applicant, or petitioning organization, should fill out Form I-360, bearing in mind that several sections -- those applicable to other categories of applicants -- can be left blank. In Part 4, you will need to designate a U.S. consulate for processing and interviewing, preferably in the immigrant's home country unless he or she is a resident elsewhere. Fill this in even if the immigrant lives in the U.S., as a backup -- not everyone who lives in the U.S. is eligible to complete the green card application process there (called "adjusting status"). Also be sure to fill out Part 9, listing your spouse and children, regardless of which category you are applying in. They may be able to get green cards at the same time as you -- or if not, you may be able to petition for them later (which will be very difficult if you did not reveal their existence in this application).
Then you will need to send the form to USCIS, following the instructions on its website. Along with the form, you must submit the proper supporting documents, as also described by USCIS, in the downloadable instructions to this form.
In some, but not all cases, a fee is required with your petition ($435 as of 2017, but check for the latest on the USCIS website). If you are among the categories of applicants who does not owe a fee with the application (Amerasians; self-petitioning battered or abused spouses, parents, or children; Special Immigrant Juveniles; or Iraqi or Afghan Nationals who worked for or on behalf of the U.S. government), you can submit your I-360 petition by email rather than regular mail -- but you will have to know how to turn your documents into pdfs to send them electronically, as well.
If using the mail, make a complete copy of everything you send before you send it (in case it gets lost -- this is not uncommon), and do not send original documents unless you do not want them back.
Once your petition is approved, you can file an application for your permanent resident status. The procedure for issuing the green card is different if conducted from abroad as opposed to within the United States.
The documents needed to support your petition will vary depending on the category of special immigrant for which you are applying. See "Process to Get a Green Card Under VAWA" for details on what documents battered spouses and children should provide.