Are there rules governing the officers of a PTO organization? Does the President have to have a student attending the school or could it be a person of the community?

PTO Chair - There are a lot of different opinions on this particular subject. PTOs write their own bylaws, typically using Robert's Rules of Order as a parliamentary authority.

If your group is already established, you should have bylaws or (as Phil has) a charter that clearly defines membership rules and the roles of each officer.

If you don't have bylaws or a charter, now is definitely the time to draft some.

Like CrewChief said, check your by-laws. Ours state:
Membership
1. Any individual who subscribes to the objectives and policies of this organization may become a member of this organization, subject only to compliance with the provisions of the bylaws. Membership in this organization shall be available without regards to race, color, creed, national origin or sex.
2. The organization shall conduct an annual enrollment of members but persons may be admitted to membership at any time.
3. All parents of students at East/West Elementary are eligible to participate in the business meetings or to serve in any of its elective or appointive positions.
4. All parents of students at East/West Elementary, all teachers at East/West Elementary, and any students or persons with a vested interest of students are eligible for membership with full voting rights.

Hope this helps.