Located in the central part of the county, it borders the following townships: Two municipalities are located in Lafayette Township: the village of Chippewa Lake in the southeast, and part of the city of Medina, the county seat of Medina County, in the northeast.

Statewide, the only other Lafayette Township is located in Coshocton County.

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, 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.

As of 2023 the board was composed of Ty Fullerton, Chris O'Neil, and Marty Warchola, and the fiscal officer is Laura Ruebensaal.