This term refers to a procedure by which someone can file a law suit even if they do not have the money to pay the filing fees that usually go with filing a law suit.
This does not mean that the government pays for the lawyer (the government is only required to pay for a lawyer in criminal cases). What it does mean is that the petitioner does not have to pay the usual fees.
A lot of petitions come to the Supreme Court and other appellate courts in this way. Most of those come from prisoners who are trying to appeal something about their convictions.