Located in the northeastern part of the county, it is bordered by the following townships: Mason County, West Virginia, lies across the Ohio River to the east.

It is located upstream of four of Gallia County's five other Ohio River townships.

No municipalities are located in Addison Township.

The township is governed by a three-member board of trustees, who are elected in November of odd-numbered years to a four-year term beginning on the following January 1.

There is also an elected township fiscal officer,[4] who serves a four-year term beginning on April 1 of the year after the election, which is held in November of the year before the presidential election.