Filing a petition is the process through which most civil court cases begin. In order for a petition to be heard by a judge in court, a notice of summons must be properly served to the opposing party.
Generally speaking, petitions must be filed electronically through the Clerk of Court's website – see here. You may qualify for an exemption if you can't file online.
The person that files the petition is the petitioner. The person receiving the petition is the respondent, and will be required to file an appearance within 30 days of receiving the notice.
A petition can be either filed by the petitioner's attorney, or by the petitioner themselves if they plan to represent themselves as their own attorney (pro se) in court.
The filing fee for a petition depends on the case, but is typically $211 USD for family law matters. You may also have to pay a fee for service of process.
If you believe you cannot afford to pay the filing fee, you may be able to qualify for a fee waiver.