Land Status on Reservations:
Tribal Trust Land: Land owned by a Tribe, placed in Trust by the US Government. All Tribal Trust Land is considered property of the Tribe, but the BIA manages it for the Tribe, with input from the Tribe. Tribal Trust Land is Tax Exempt.
Tribal Allotted Land: Allotted Land is land owned by an individual Tribal member or his family (heirs). This is land "given" to individuals for the purpose of farming and ranching among other things. This land is also in Trust. The government (BIA) has the responsibility to manage all income from this land for the various ownership heirs. This land is also Tax Exempt.
Fee Land: Also known as deeded land, is property owned by people living with in the exterior boundaries of the reservation that hold title to their property. Fee land is not held in Trust by the BIA. Fee Land is subject to taxes from the county where it is located. Most non-Indian landowners living on the reservation own fee land.
There are several other sub types of property, but these three are the major types of reservation land.
Most Tribes have their own "Law and Order" code. These are a set of laws like any other city, county or state may have. Most reservations are not obligated to follow State law, but many have similar laws to the States they are located in.
Most Tribal Law deals with misdemeanor crime that happens within their boundaries. On most Reservations, felonious crime is tried by the Federal DOJ in federal court. Some reservations have their own police force, but others still rely on the BIA to provide police officers.
State and local police jurisdictions also sometimes have MOU's with Tribal law enforcement. These MOU's deal with non-Indians who commit crime on reservations. Complicating things, Federal law, State law and Tribal law sometimes apply to non-Tribal folks within reservation boundaries.
Reservations can make their own gun laws, and often do. Most times, though, if persons remain on State or Federal Highways, the law that governs the rest of the highway off reservation is what one should follow.
Finally, Reservation land IS NOT public property! You can get into big time trouble for trespassing on Reservation property!