Questions regarding ELISA
ELISA is Enzyme-Linked ImmunoSorbent Assay (capitalisation to point out source of acronym).

You can read about the various ways to do an ELISA here.

In your case this was probably an indirect ELISA. All ELISA techniques make use of the fact that proteins stick to plastic. In your case the protein components of the saliva, including the virus (the viral coat is proteinaceous) will stick to the wells. After rinsing out the sample the wells are treated with another protein (usually bovine serum albumin (BSA)) to completely block all available protein binding sites. At this stage these wells will no longer bind protein non-specifically, but will bind anti-EBV Ig because of the presence of the EBV particles bound to the plastic. This bound anti-EBV can then be detected with a secondary antibody. The protein complexes will build as follows:

plastic/EBV/rabbit anti-EBV/goat anti-rabbit IgG-HRP conjugate

It would also be possible to do this experiment as a sandwich ELISA. In this format the wells would have been pretreated with anti-EBV serum (or purified anti-EBV IgG) then blocked with BSA. At this stage these wells will no longer bind protein non-specifically, but will bind EBV because of the presence of the EBV-specific antibody. Once samples have been incubated in these wells (so that any EBV can bind to the antibody coating the wells) another antibody (raised in a different animal) is added, and this will bind to the EBV that has been captured by the original antibody that coats the wells. This latter antibody is then detected with a conjugate. The protein complexes will build as follows:

plastic/mouse anti-EBV/EBV/rabbit anti-EBV/goat anti-rabbit IgG-HRP conjugate

Your description of how to get the antibodies is basically correct, except that the primary antibody will probably have been raised in rabbits or mice, and certainly not in humans.

Added later: I didn't read the first question carefully enough. An ELISA is an assay because it measures something. It is based on a binding reaction mediated by an antibody-antigen interaction (immunosorbent). It is enzyme-linked because the secondary antibody is present as an antibody-enzyme conjugate. It is the amount of this conjugate that remains bound at the final step which is what is measured - the assay is set up so that the amount of this enzyme is a measure of the amount of the original molecule that is being assayed.