Local government refers to local councils, which are also sometimes called local authorities.
The country is split up into different areas and each area has a council that takes care of local issues.
Councils control things like council housing, rubbish collection, recycling, parking, transport, roads and footpaths, parks, leisure centres and libraries.
Councillors are responsible for making decisions on behalf of the local community. They come up with policies and officers are responsible for making them happen. This is similar to central government where ministers come up with policies and civil servants are responsible for making them happen.