The local English village of the period was governed at the parish level, which was traceable to the ecclesiastical parishes which were based around the local church and the area it surrounded, most often based around the local manorial area. So, by the early 17th century you have this confluence of local administration for the local church and the local civil activities. Now parishes could vary widely in size from a sparsely populated area of lade with a hundred inhabitants, right through to a large town with several thousands. The average in the sense you are asking is probably a few hundred inhabitants spread over the area covered by the local church, this may well include more than one village although the largest village would probably hold the local church, with the rest of the populations scattered amongst smallholdings and small hamlets nearby.

The people who governed locally would actually be locals themselves, usually drawn from the wealthiest householders and employers in the area. They would be elected or selected from amongst their peers on a rotating basis. Primarily in this period there would be four offices, with a varying number of officers for each role dependent on the size and needs of the parish. These were:

Parish Constables: Responsible for local administration of the militia, apprehension of criminals, escorting vagrants and paupers out of the parish, licensing of inns, serving legal orders from higher authority. It was also the constables responsible for collecting money when a tax was called, and for locating men for the press when one was levied for the army.

Churchwardens: Responsible for the upkeep and maintenance (and associated fund raising) for the local church, its land, etc. They were also often the local Overseers of the Poor or at least worked heavily alongside them. After the Reformation, with the collapse of the support for the poor historically administered through the local Church tithe, the Churchwarden was responsible for assessing and gathering in necessary funds to support those in the local community reliant on poor relief.

Surveyors: The final elected local officer was the Surveyor of the Highways, who was responsible for maintenance and repair of the roads which ran through the parish, and which the local householders were obliged to not only fund, but also for six days a year to provide physical labour for their upkeep

There would also have been often some form of Parish Clerk, usually salaried, responsible for all the little jobs necessary to assist the officers, not least filling out the records of the officers if the elected official was illiterate themselves.

Note: There may have been different names for the offices across the country. Constables could be called Parish Constable, Petty Constable (both to differentiate from High Constable), Bursholder, Thirdburroughs, etc.,

Being elected to these officers was seen as rather a burden to the local householders, making them responsible for their respective officers for the length of a term which could be one to two years usually. Refusing the office could only be done with payment of a fine, and the holder could be dismissed from office by higher county authority if they were deemed ‘insufficient persons’, that is not enough standing in the local community to perform the office.

If these officers failed, or neglected their responsibilities, which were many, they themselves were fined for their failings. Other punishments could also occur if higher county or shire administration felt the parish had failed in its administration. For instance, a petty constable that failed to impress the requisite number of men allocated to be conscripted in his area, could be liable to be impressed himself in order to make up the numbers. This risk of fines and punishments goes someway to explain why the responsibility of the office was avoided when possible.

Bear in mind that in the period, the only people who were taxed, or who paid rates, as well as those who provided money for the poor and the needy, as well as the funds required by the parish officers were all from the same (small) group of householders and businesspeople liable for selection as parish officers. It would not be unusual for an individual in the community to have to perform various of these offices in their lifetime.

How one performed an officer you were selected for without any formal training or prior experience was a problem. Often you were reliant on guidance from previous officer holders, but the work was considered important enough that printed guides to being one of these officers did exist, for example Meriton, George, A guide for constables, churchwardens, overseers of the poor, surveyors of the highways, treasurers of the county stock, masters of the house of correction, bayliffs of mannours, toll-takers in fairs &c. a treatise briefly shewing the extent and latitude of the several offices, with the power of the officers herein, both by common law and statute (London:1669)

I am happy to answer any follow up questions if I can.