ELISA is a fantastically versatile tool that is used for diagnosing viral infections and the like, but technically speaking, it can be used to detect and quantify any antigen or antibody that one may so desire to look for. It stands for “Enzyme Linked Immunosorbent Assay” and that means a test which uses an immunosorbent (a substance that either is or reacts with an immune compound to bind to it) linked to an enzyme.
Before moving on, it is important to know that antibodies are chemical substances that the immune system makes which bind in a highly specific manner to other molecules, which are called antigens. (This definition is tautological, an antigen is a substance that binds to an antibody and a substance that binds specifically to an antibody is an antigen) , it is also important to note that antibodies are proteins, and may be suitably modified and linked to other proteins using good old chemistry. The process of doing this is called conjugation and this is how one produces the enzyme linked immunosorbents used in the assay.
Depending on whether one intends to detect antibodies or antigens, the setup used for ELISA may vary. In case we are looking for antibodies, we can immobilize the antigen onto a substrate, in case we are looking for antigens we can immobilize antibodies onto the substrate.
ELISA is usually carried out on strips of cellulose acetate (this is called Dot ELISA) or in microtitre plates.
 Depending on whether one is looking for antigens or antibodies, antibodies or antigens, respectively, are chemically bound to the surface of wells/test strips. While scientists have to do this while creating their own assays, those that are used for diagnosis come with those molecules already bound.
 Areas of the well that haven’t been coated with the antigen/antibody are then blocked using a blocking buffer, the well is now ready to use.
 The sample is then added and if there are antibodies present to the bound antigens they will bind to the well, too. These wells are then washed to remove any unbound antibodies.
 A secondary antibody which is covalently modified and linked to an enzyme is then added, this antibody binds to any antibodies that have bound to antigens in step  , finally, an enzyme substrate is added the enzyme converts this to a product which can be measured using optical methods such as colorimetry. This enables quantification as well as detection.
Sandwich ELISA is used when we are looking for antigens, in this case a primary, unlabelled antibody is bound to the wells, if there are antigens they will bind to these antibodies, a second set of labelled antibodies is then introduced. These antibodies bind to the antigen bound to the primary antibody. Again a substrate for the enzyme is added and this produces an optically measurable signal.
Some setups use an unlabelled antibody to bind to the antigen and a labelled antibody to bind to the antibody that is bound to the antigen. Again, it is one of the many possible configurations that may be employed.Please click on the image to bring up a larger version.
As already mentioned, they can be used for detection and quantification of either antibodies or antigens and this immediately has diagnostic implications. It can also be used in some circumstances to quantify the effect of drugs on the expression of particular proteins (in this case by analysing the quantity of bindable antigen) and so on and so forth. It is not only used clinically for diagnosing viral infections but also for things like pregnancy tests, where assays are geared towards the detection of proteins like HCG (Human Chorionic Gonadotrophin)
You may watch videos of various methods of ELISA being employed and summarized here
That is all from me about ELISA at this juncture. Happy reading.