1 An area that includes a large city and suburbs
Typically, a tri-state area is a region that includes a large central city and its residential suburbs or exurbs that extend into three states.
2 Simply an area that extends to 3 states
A tri-state region is a region of three states. It also depends on the states that you live in. It generally means some area or region inhabited by people that extends to three political states in the United States of America.
3 Cincinnati Metropolitan Area
A good example includes the Cincinnati metropolitan area. The city itself is part of Ohio state but includes counties from other states such as Boone County, Kentucky, and Franklin County, Indiana in the metropolitan area. These incidents happen because the central city could be located near the border of the state, and settlements grow as part of the city, crossing over to the nearby state(s).
4 New Jersey
The typical tri-state area for New Jersey is considered to be New Jersey, New York, and Pennsylvania.
5 North Eastern New Jersey
The tri-state is considered to be New Jersey, New York, and Connecticut.
6 New York
The suburbs and exurbs are in the states of New York, Connecticut, and New Jersey.
In the case of Philadelphia, the suburbs and exurbs are in the states of Pennsylvania, New Jersey, and Delaware.
In the case of Pittsburgh, the states would be Pennsylvania, Ohio, and West Virginia. (See What exactly does ‘processed through ISC New York’ mean on USPS tracking?)
9 Some other tri-state areas are
Massachusetts, New Hampshire, and Rhode Island (Boston)
New York, New Jersey, and Connecticut (New York City)
Pennsylvania, New Jersey, and Delaware (Philadelphia)
Ohio, Kentucky, and Indiana (Cincinnati)
Illinois, Indiana, and Wisconsin (Chicago)
Iowa, Nebraska, and South Dakota (Sioux City)
The term is also occasionally used for the District of Columbia, Maryland, and Virginia (Washington, DC).