By districts I'm assuming you're speaking of congressional districts, which are set up as local representation to Washington. These districts do nothing actually as far as local government. Counties on the other hand are local. I think the confusing part here is how the two governments break down.

National breaks down- National > state > District

State (local) government - State > County > City

What do you mean by districts? Something other than District of Columbia?

You're confusing your terms. Let's lay down some basic terminology.

States: The basic division of land in the U.S., and one that predates independence. There are 50 of them, and their borders cannot be changed without the state's consent. In this vein, Maine used to be part of Massachusetts, and the Oklahoma panhandle used to be part of Texas. (West Virginia was split off from Virginia during the Civil War in 1863 and it's a special case.) Under the 10th Amendment to the U.S. Constitution, state governments have all government powers not reserved to the federal government. It would be legal for state governments to provide all state government services directly, but it's not practical to do so. State governments just aren't responsive enough to handle day-to-day government functions like zoning.

Counties: Units of local government that exist under state law. Like I said before, state governments generally aren't responsive enough to handle local government issues, which is why every state has delegated government to varying degrees to counties. (The oldest counties in the United States date to the 1600s.) The various states have taken widely divergent approaches to county government as communications and government responsibilities have expanded. The county governments are instrumentalities of the state government, however, and the balance of responsibility can be shifted at any time by the state legislature.

At one extreme is Rhode Island, where county government has been abolished entirely, and the county system is used for administrative purposes only, like for the census. (Services previously provided by the counties were either integrated into the state, or else devolved to the city level.) At the other extreme are the southern states like Virginia, where county government is a major unit of local government and most public services are provided through the county.

Consolidated City-County: This is a fairly common arrangement where a city government has merged with the county government. This happens most often in major cities, like San Francisco (San Francisco County), New Orleans (Orleans Parish), and Philadelphia (Philadelphia County). A similar form is the independent city, which technically isn't a consolidated city-county, but acts as one in all but name. This is the case in Baltimore, St. Louis, and Richmond, VA.

Districts: The one universal district type in the US is the Congressional district, which exists to provide representation in Congress. This divides every state in the Union equally by population. Congressional districts exist solely for the purpose of representation, and don't provide any other services. (They change after every Census, as the population shifts.) Congressional districts actually postdate the existence of state, city and local government, as they only came into existence in 1789 after the ratification of the US constitution.

There are also lots of other types of "districts" that provide government services, usually because often ordinary government boundaries don't match what's needed. Let's use the East Bay region, California, as an example. There, because the county lines make it difficult for individual counties to provide services, the East Bay Municipal Utility District provides water and sewer service across both counties, the Bay Area Rapid Transit District provides subway service, and the East Bay Regional Park District takes care of the much of the parkland.

They each have different functions. Electoral districts to give them their full name are part of the electoral system. A country on the other hand is a part of the administrative system of the US.

States and counties are part of state government, a lot of the day to day running. Districts are a part of the federal electoral system and have no real administrative functions over day to day life. A congressman can't come into their district and start saying "This is how schools and state jails should be run!" because they have no authority.

So counties are local government, districts are federal government. Hope that helps.

The designation of county actually has its origins in the Roman Empire, with the term comitatus, which meant a body of companions or retinue, typically of a military commander who was about to enter into battle. By extension, the term comes developed into a secular title granted to trusted officials of the Imperial Curia or Court. Over a course of time the Comes or Comites became important officials of the Empire, typically playing important roles in the army or civil service.

By the 8th century, the term became associated with the figure of the “count”, and had developed into a hereditary rather than an appointed office. The “county” (or the equivalent in various Latinate languages) was the territorial jurisdiction of a count. This term was imported into England with the Norman Conquest, as the Anglo-Norman was used as the equivalent of the Old English Earl, denoting a person with authority over one or more shires. Over a course of time, the English came to prefer the term “county” over “shire”, although the latter persists in the names of various counties (Cheshire, Derbyshire, Hampshire, etc.).

Note that the county did not merely refer to the territory, but also meant the convention or court that was called into session by a sheriff to conduct the business of the shire.

So when the English began to establish colonies in North America, they brought this unit of administration with them. While colonial Virginia and New England developed different municipal jurisdictions (towns, hundreds, cities, etc.), they both divided their colonies into counties – largely because of the importance of this unit in administering local justice, and in recording property rights in land.

At the time of the Revolution, every colony had divided its jurisdictions into counties, administered through county courts that not only heard legal cases but also conducted most of the business of local administration, from granting tavern licenses to repairing bridges.

So in point of fact, counties actually precede states and electoral districts as administrative units in the United States.