There are different categories of maintained schools as well as academies and free schools, which, though publicly funded, are independent schools. If the school is a community school or is voluntary-controlled, the local authority (LA) decides on the admission arrangements and applications are sent to the LA for consideration.
For foundation and voluntary-aided schools, academies and free schools, it is the school’s governing body or academy trust that decides the admission arrangements and which children best meet its oversubscription criteria if it has more applicants than places.
Whether you apply direct to the school or via the LA depends on whether the LA coordinates admissions. During the normal admissions round when children are of the age to start school (ie Reception for primary school and year 7 for secondary school) the local authority (LA) coordinates the admissions applications for all schools in its area. At other times of the year, the process will depend on the type of school. For community and voluntary-controlled schools you would need to apply to your LA.
Academies, foundation and voluntary-aided schools may organise their own admissions, so the LA may tell you to apply directly to the school. Contact your LA for more details.