How are local councils actually structured, and what do they do?
Differently in different places — which is the honest starting point, because the UK has several parallel structures and knowing which applies to you determines who is responsible for what.
The structures:
Two-tier areas, where a county council and district, borough or city councils split responsibilities. County handles education, social care, transport and waste disposal; district handles housing, planning applications, waste collection, licensing and council tax collection. This split is the source of most "that's not us" responses.
Unitary authorities, where a single council does everything. Increasingly common through reorganisation.
London boroughs, with the Greater London Authority above them.
Combined authorities and metro mayors, covering transport, skills and strategic planning across several councils.
Parish, town and community councils, the smallest tier, with limited powers over local amenities, allotments, some planning consultation and local facilities — and a precept they add to council tax.
What councils actually deliver. The largest items by spending are adult social care and children's services, which together consume the majority of many councils' budgets — a fact that surprises residents who associate councils with bins and potholes, and which explains why other services are squeezed.
Then: education support, highways, waste, libraries, leisure, planning, licensing, environmental health, trading standards, housing and homelessness, and public health.
How decisions get made: elected councillors; a cabinet or committee system; full council for budgets and major policy; planning and licensing committees, which are quasi-judicial and must decide on defined grounds; and officers, who do the work and advise.
How councils are funded, which explains a great deal: council tax, retained business rates, government grants, and fees and charges — with substantial real-terms reductions in central funding over the past decade and statutory duties that cannot be reduced, which is why several councils have issued section 114 notices effectively declaring they cannot balance their budgets.
How to get something done: report through the correct council's system; escalate to a ward councillor, which is frequently more effective than the complaints process; use the formal complaints procedure; then the Local Government and Social Care Ombudsman.
General information, not legal advice.