It is simply conjugation of enzyme to pry antibody or secondary antibody; that is, how detection conjugate is bound. With pry Ab, it gives visible reaction directly thus called DIRECT method, while with secondary Ab telling us that the pry Ab is not directly unveiling the type of the antigen on binding but rather showing this through secondary Ab and therefore INDIRECTLY.
Direct ELISA: the antigen is then detected by an antibody directly conjugated to an enzyme, such as HRP.
Indirect ELISA: detection is a two-step process. First, an unlabeled primary antibody binds to the specific antigen. Second, an enzyme-conjugated secondary antibody that is directed against the host species of the primary antibody is applied.
It is simply conjugation of enzyme to pry antibody or secondary antibody; that is, how detection conjugate is bound. With pry Ab, it gives visible reaction directly thus called DIRECT method, while with secondary Ab telling us that the pry Ab is not directly unveiling the type of the antigen on binding but rather showing this through secondary Ab and therefore INDIRECTLY.
all the upper explanations are right, however, there is no strict nomenclature in ELISAs! in the upper cases direct ELISA means, that a primary (= target binding) antibody is labeled and an indirect ELISA you use a secondary labeled antibody.
But these terms are also used sometimes for competitive ELISAs, there it is a bit different. Competitive ELISAs are used for small molecule detection. For a direct competitive ELISA the primary antibody is immobilized on the solid surface (e. g. on a microtiter plate) and use use a Tracer, which is a enzymatic labeled antigen as competitor. This is a conjugate from your small molecule and e.g. HRP. For the indirect competitive ELISA the antigen is immobilized on the surfaces of the Microtiter plate and the competitive reaction happens between the immobilized antigen (which is usually a conjugate between your analyte and the small molecule) and the analyte in solution. So the primary Antibody is in solution.
So if you are working with a competitive ELISA, the appellation is different.