A reporter gene is a gene whose protein product is easily detectable and/or measurable, in itself (e.g. using an antibody) or by a product of its activity.
Widely used reporter genes include Chloramphenicol Acetyl Transferase (CAT), whose product has an "easily" detectable enzymatic activity, lacZ, easily detectable by an histochemical reaction, luciferase, eGFP, etc.
A reporter gene should encode for a protein whose activity is zero in the cell/animal worked on, so that the background is theoretically zero.
Reporter genes are used to study the structure function of the regulatory sequence of a given genes, when the expression of the natural product of that gene is difficult to detect, etc.
Practically, the coding portion of the gene studies is replaced with that of the reporter gene: the regulatory sequences will now affect the expression of the reporter gene. Construction containing deletions in the promoter sequences are transfected into a suitable cell, and the results of these deletions are followed by the expression of the reporter gene.