In some states it breaks down that the State Police are rural law enforcement (areas which may not be covered by other agencies), but mostly they are for hwy traffic enforcement and accident related issues.
County Police (and larger incorporated cities also fall into this category) are used for law enforcement purposes dealing with traffic, criminal, disputes, etc.
Sheriff are used for court related issues, summons, evictions, warrants, jail duties, things of that nature
The problem is that unlike most countries the USA is very diverse in the way it proceeds with functions, in some places EMS is under the umbrella of fire, in other places they subcontract it out to AMR and companies like that... You can't really cover the whole of the USA with one specific setup, usually though larger cities have their own PD which generally covers the incorporated territory of the cities, followed by a county around said city that is enforced usually by a county police, as you get further towards rural areas it becomes a bit more confusing as in some municipalities its sheriff in others it police, although in general all share the same "powers" with regards to their positions. The difference is mostly just in the label slapped onto them, some places have transportation/transit police that are for traffic enforcement on toll roads, others do not but in the end the police duties remain the same.
The exception to that generally is the "auxiliary" police, they generally are not as equipped on powers under their command, generally used to augment the primary police force in times of emergency such as wildfires, snow storms, anything that may cause a state of emergency or require massive response, sometimes they are brought out merely for there being a concert of some other venue which requires additional manpower but only on a temp basis.