[1] Similar to a city council, a board of supervisors has legislative, executive, and quasi-judicial powers.
The important difference is that a county is an administrative division of a state, whereas a city is a municipal corporation; thus, counties implement and, as necessary, refine the local application of state law and public policy, while cities produce and implement their own local laws and public policy (subject to the overriding authority of state law).
Boards may pass and repeal laws, generally called ordinances.
Both Michigan and New York changed how they elected county boards by dividing counties into single member districts, drawn so that each district has more or less the same sized population or with each township's vote weighted by population, under order from the Warren Court (see Reynolds v. Sims).
Those that retained the old boards of supervisors after the 1960s assigned each member a proportional vote based on the population represented.
In Pennsylvania, "Board of Supervisors" is the name of the body governing townships of the second class that have not adopted a home rule charter.
By default, a Pennsylvania township board of supervisors consists of three members, elected at large in odd-numbered years to staggered six-year terms.