Celoron is located in the western part of the town of Ellicott at 42°6′24″N 79°16′53″W / 42.10667°N 79.28139°W / 42.10667; -79.28139 (42.106778, -79.281414), on the south shore of Chautauqua Lake. It is bordered to the east by the city of Jamestown and to the south and west by unincorporated parts of the town of Ellicott.
According to the United States Census Bureau, the village has a total area of 1.9 square kilometres (0.73 sq mi), all of it land.
As of the census of 2000, there were 1,295 people, 526 households, and 357 families residing in the village. The population density was 1,741.0 people per square mile (675.7/km²). There were 567 housing units at an average density of 762.3 per square mile (295.8/km²). The racial makeup of the village was 94.90% White, 0.77% African American, 0.46% Native American, 1.54% Asian, 0.54% from other races, and 1.78% from two or more races. Hispanic or Latino of any race were 2.16% of the population.
There were 526 households out of which 30.4% had children under the age of 18 living with them, 50.0% were married couples living together, 11.8% had a female householder with no husband present, and 32.1% were non-families. 26.8% of all households were made up of individuals and 11.2% had someone living alone who was 65 years of age or older. The average household size was 2.39 and the average family size was 2.83.
In the village, the population was spread out with 23.8% under the age of 18, 7.3% from 18 to 24, 28.9% from 25 to 44, 25.7% from 45 to 64, and 14.3% who were 65 years of age or older. The median age was 38 years. For every 100 females, there were 95.9 males. For every 100 females age 18 and over, there were 93.5 males.
The median income for a household in the village was $31,544, and the median income for a family was $33,333. Males had a median income of $30,980 versus $21,719 for females. The per capita income for the village was $15,098. About 8.9% of families and 13.2% of the population were below the poverty line, including 13.6% of those under age 18 and 5.4% of those age 65 or over.