A private airport is any airport that is not open to the public. They can be airports where there are memberships sold to specific individuals, or airports that belong to private communities. Private airports can also be airports that are owned and operated by private individuals and are not open to anyone but those who own them. However, access to a private airport is not completely out of the question if you have the pre-approval of the owner or operator of that airport. Some private airports just require the filing of a flight plan with air traffic control as well as the local flight service to gain access, but this is not the case most of the time.
A large majority of private airports are grass or dirt strip fields without services or facilities for individuals looking for lodging, fuel, or tie-down slots. In the event of an emergency, landing at a private airport is approved without any prior approval and should be done if landing anywhere else compromises the safety of the aircraft, crew, passengers, or cargo. If looking on a sectional, a private airport is denoted by a small magenta circle with an "R" inside of it.
Due to the fact that the airport is private, the owner, and operator are, in most cases, responsible for all costs for maintenance, repair, and upkeep of that airport. The FAA does make exceptions for private airports that make exceptions and funds can be allocated to individuals who will maintain the airport up to FAA standards, such as paved runways. This is why the most private airports are grass or dirt strips.
Certain private airports, like those for small housing communities, usually for the selectively rich, might require fees to maintain the airport and thus find ways around conforming to FAA standards like grass and dirt strips. However, since the cost of pavement is high, private airport owners reluctantly conform.