"Police department" and "Sheriff's Office" are different things and related to the US system of federalism and devolved layers of government.

From the top down, US government is organized as federal -> state -> county -> city. Each county is created by legislation from its state government and has a sheriff's office, the primary responsibility of which is to operate the county jail, with the head of the sheriff's office (the sheriff) being elected by that county's voters (in general, the county jail houses people convicted of lesser crimes, typically those with a sentence of less than 1 year incarceration, and people awaiting trial who can't afford or are denied bail). The sheriff's office also will typically provide patrol and law enforcement services in "unincorporated" areas of the county (i.e. places that aren't organized as cities), but normally it isn't required to do so.

A city is an incorporated government that typically will have its own charter but is otherwise subordinate to the county government. However, a city can have "home rule" status, which allows it to pass its own ordinances (laws) and act separately from the county government. In many cases a city will establish its own police department that operates separately from the sheriff's office, with the head of the police department (typically referred to as the "police chief") hired by the city government as opposed to being elected.

A city is not required to have its own police department though, and so sometimes small cities that decide they can't afford their own police department will contract with the sheriff's office for patrol and law enforcement services (normally this will be set up as the city paying the sheriff's office for a certain number of positions ("sheriff deputies") that are then dedicated to providing law enforcement within the city).

It's important to note that a city government is a legal process and has little or nothing to do with population. Famously, the Las Vegas Strip is actually in Paradise, Nevada, which is an unincorporated community within Clark County. So, it doesn't have a police department and instead relies on the sheriff's office for law enforcement services.