Direct ELISA is based on the binding of a labeled antibody with the antigen of interest already coated in the well. This is not very widely used. However, in the indirect ELISA, a labeled secondary antibody for detection. This antibody has an affinity for the primary antibody and is highly sensitive assay. This is also known as Sandwich ELISA.
Actually, a sandwich ELISA is not the same thing as an indirect ELISA. In a sandwich ELISA, the plate is coated with a capture antibody (antibody 1) that recognizes the antigen to capture it. A second antibody (antibody 2) that recognizes a different epitope on the antigen is then used as the first step to detect it. At this stage, indirect detection, as described by Mayank Garg, can be used to detect antibody 2.
Indirect detection in a sandwich ELISA can make use of yet another antibody (antibody 3) that binds to antibody 2 but not antibody 1. This is usually accomplished by having antibody 1 and antibody 2 be from different species, and antibody 3 recognizes the species of antibody 2. Antibody 3 is conjugated to an enzyme that is used for detection, normally either horseradish peroxidase or alkaline phosphatase.
An alternative approach is to use a biotinylated antibody 2. It is detected with streptavidin that is conjugated to an enzyme.
Indirect detection can also be done with simpler ELISAs that are not sandwich ELISAs. The antigen is bound directly to the plate surface, so there is no capture antibody.This is a less sensitive method, but is much simpler to develop than a sandwich ELISA.