Most but not all of the New Forest belongs to the Crown, which broadly means that it is owned by the government on behalf of all of us. Other parts of the open Forest that are not owned by the Crown are called ‘Adjacent Commons’. There are still practical differences in the way that the Crown land and the adjacent Commons are managed.
For example, Forestry England looks after the parts of the open Forest that are owned by the Crown, and their bylaws apply there. In other parts of the Forest you might see signs showing that they are owned and looked after by the National Trust, for example.