In a university or college, adjunct faculty are faculty that are adjunct. That is, they are faculty that teach as an addition or extension of the normal faculty of the institution.
This translates into many different things, and the evolution of "adjunct faculty" has changed to the point where the literal meaning of the term does not describe who they are, or what they do. The original intention of having adjunct faculty so was that professional who were still practicing their occupations could help teach in higher education, giving them a chance to share their real world experience. For example, a practicing or semi-retired accountant could help in teaching accounting. Since they were otherwise occupied, the pay for teaching part time was a relatively minor enticement.
In some places, this is probably what the system still means. Higher education in the United States encompasses a broad spectrum of resources and purposes. However, in the community college system, where I currently teach, "adjunct faculty" has taken on a different meaning. As opposed to its stated title of being an addition to the regular faculty, the "adjunct faculty" make up much, or most of, the faculty in many community colleges. And rather than being professionals who are otherwise employed and teach on the side, they are mostly teachers who are not teaching full time due to economics. Adjunct faculty are faculty that are kept on a contract basis, teaching from term to term as enrollment dictates, and are not given benefits. Depending on the institution, they would also usually not have personal office space or keep office hours.
Keeping faculty on as adjuncts is one way that community colleges are able to keep costs down, and keep community colleges one of the few options available for students who don't want to go into debt. However, it does have a downside, the biggest of which is that since they are kept as peripheral members of the institution, adjunct faculty are not always able to devote the time to their students that more constant faculty would. Also, for the adjunct faculty themselves, the pay is not very good for the level of education involved. I personally have mixed feelings about this: I am very happy to be making (around) 25 dollars an hour, but I am not so happy to be working six hours a week. Adjunct teaching is a fun and high-paying job--- as long as you already have a job.
I am no expert on higher education economics (although I should be), but I see why a college would prefer to use adjuncts. The care and feeding of full time faculty grows prohibitive. Offices, personal assistances, benefits, sabbaticals, and conferences really add up quickly. For example, it is quite conceivable that the cost of sending a professor to a conference, including airfare and lodging, could run to several thousand dollars, equal to what an adjunct faculty member might make in a year. While the system of hiring adjunct faculty might only be a band-aid to deal with the spiraling cost of higher education, it is currently one of the only games in town. And one that I personally must play, for the foreseeable future.