In a legal context, a "petition" is the document a plaintiff files to start a law suit (also known as a "complaint"). To get a copy of the petition filed in a particular case you can (a) get it online, if it is available (see the "Docket Sheets" entry), (b) call the court to see if they will copy and mail it to you, (c) pick it up yourself or (d) get someone to pick it up for you which, for courts in other cities, usually means hiring a document retrieval (see "Document Retrieval Services").
Certiorari Petitions: Petitions for Certiorari are filed with a higher-level court to ask that the court review a lower court's decision. Generally, you can find out whether cert. has been granted in a particular case by calling the court, and you can get copies of cert. petitions the same way you get copies of other petitions (see above). Exception: For cert. petitions filed with the U.S. Supreme Court, see "United States Supreme Court."