A feasibility study is a formal report that documents decisions concluding in a choice of one from two or more alternatives. It establishes possibilities, evaluates economics, and forces reflections on the perceptions of a decision (social, political). It begins, of course, with a consideration of the audience that will read the study and your purpose in writing to that audience, as always.
The feasibility study is an important tool in engineering and management because it formalizes much of the openness of a brainstorming process (it forces you to think of novel solutions), and it encourages you to think in quantifiable terms about real dimensions to the problem you are trying to solve. The phrase "systems approach," used below, refers to a number of steps, taken as part of a formal study, that begin with problem definition (via a feasibility study) and progress through modeling, selection and construction of a single solution, and follow-up evaluation of the real-world system.
In the feasibility study, the problem is carefully described, and one or more potential solutions to the problem are considered in some depth. Later decisions about what particular steps to take to actually effect a solution are based upon the results of the feasibility study. Often, the feasibility study -- in the form of a grant proposal or an impact assessment statement -- provides the basis for action by funding agencies. A well done feasibility study should tell its author whether further action is desirable, and should be able to convince others as well.