"To run the council" means to manage and make decisions for a group of people who oversee the activities and policies of a government organization at the local level. Full definition
What would happen is all 130 Labour run councils refused to set «legal» budgets? (leftfutures.org)
Labour were running the council on a minority basis before this election after a decade - long spell when no one party has had an overall majority. (itv.com)
From 2004, however, no party had a governing majority and a Conservative - Lib Dem coalition ran the council from 2004. (democraticaudit.com)