Geneva had its start in 1871 when the rival towns of Alexander and Buffalo merged. It was named after Geneva, in Switzerland. The first post office in Geneva was established in 1872.
As of the census of 2010, there were 1,293 people, 570 households, and 349 families residing in the town. The population density was 1,186.2 inhabitants per square mile (458.0/km2). There were 627 housing units at an average density of 575.2 per square mile (222.1/km2). The racial makeup of the town was 94.6% White, 0.6% African American, 0.2% Native American, 0.2% Asian, 3.2% from other races, and 1.2% from two or more races. Hispanic or Latino of any race were 5.4% of the population.
There were 570 households of which 28.8% had children under the age of 18 living with them, 45.3% were married couples living together, 11.1% had a female householder with no husband present, 4.9% had a male householder with no wife present, and 38.8% were non-families. 34.6% of all households were made up of individuals and 14.1% had someone living alone who was 65 years of age or older. The average household size was 2.27 and the average family size was 2.91.