A finishing school is traditionally a school that teaches social graces and cultural attitudes to students, presumably young women, in order that they can be proficient in social gatherings, and eventually land an eligible young man to marry. They were usually designed for women who had finished their secondary education, who didn't wish to go on to study academically. As can be imagined, schools like this were targeted at those who had both the resources, and the need to appear very culturally elite. In other words, the very wealthy, or in Europe, the aristocracy.
The finishing school, as an institution, seems to have died down, at least in civil lands. After all, women are more interested in learning job skills and getting a career of their own, rather than trying to impress a man who has a fancy job. In addition, social changes have meant that even the upper class now are not quite as concerned with things like which fork to use and where to put your pinky while drinking tea.
However, since I have had the strange academic choice of studying post-secondary education, I can confirm what most people already suspect: just about any education that people receive is meant for social education, and the college and university system is certainly no exception to this. Very few of the studies that college students go through give them direct job skills, but instead give them general information and attitudes about the world. Positively, this is meant to give students the ability to think critically, adjust to new circumstances and have a broad view of the world. Cynically, a college education is about giving a student the veneer of gentility before they go off to get a sinecure in banking or management.
To what extent a college or university can be considered a finishing school varies greatly depending on what type of college it is (state schools are less so than private schools), what the student is studying (accounting would be considered less so than literature), and where in the country it is located, (Higher Education is probably more of a class issue in New England and the Mid-Atlantic than it is in the Midwest or Mountain States). It also depends on the student's own attitude towards their education, a student dropped in a school where Greek Philosophy is covered as a matter of sophistication can choose to take the subject seriously. I would also name some schools that I think are meant to be finishing schools, but I would probably be biased, and gain some disfavor. I will let people draw their own conclusions on the purposes of the educational institutions around them.