The prayer ministry has to do with ministering to people - Believers and unbelievers - through prayer intercession and supplication. Every Believer in Christ Jesus has a level of ministry in prayer towards people - both to the Saints and the sinners. You pray for the people you preach to; and you pray to God to help this or that person in one area or the other. However there are people who have greater responsibility in ministry and assignment to pray for different kinds of people and for the Plan and Purpose of God to come to pass on this earth. Of course this does not mean that prayer is (or must be) the only ministry that the person has; because an apostle a prophet an evangelist a pastor a teacher a deacon a musician an usher and others can have a strong call in praying and interceding for people communities churches ministries regions and nations.