As the Petitioner you would the one filing the initial Petition or Motion. There is a filing fee required to be paid to the Clerk's office for the first filing. Then the other party needs to be notified of the Petition/ Motion usually through a Summons, which is served by the Sheriff and also requires a fee.
If you have been served by the Sheriff , then you are the Respondent. You must file an Appearance which requires a filing fee. You may also file an Answer or Response to the Petition/Motion.