'No party majority' is synonymous with 'no overall control' (which is probably the term that's used most often). It occurs where no party has more than 50% of the seats.
In some cases the party with the most seats runs the council as a 'minority administration' because the other parties can't find a way of getting together to form a larger group. In England, that's currently happening in six Labour-led councils, six Conservative-led councils and one LibDem-led council.
In other cases two or more different parties team up together, so that they've got more than 50% of the seats in total, to form a 'coalition administration'. There are currently three councils in England where Labour and the LibDems have formed such an alliance, one where Labour have teamed up with the Green party and one where they've partnered with Independent councillors. Conservatives have joined up with Independents to run three councils. Elsewhere in England, in one council, they've had to team up with both LibDems and Independents to gain an overall majority.
Information sourced from here:
https://www.nationalworld.com/news/politics/no-overall-control-local-equivalent-hung-parliament-explained-election-3681445