Theodore Roosevelt acquired a Bachelor of Arts degree. He graduated from Harvard in 1880, partway through the long and celebrated tenure of Charles William Eliot, who is credited with introducing concentrations to Harvard, though I have been unable to ascertain for sure the exact timeline for their introduction. I suspect it was gradual, over his long tenure. The idea of a college "major" only gained currency in this time, with the first reference to the term in a Johns Hopkins catalogue in 1879. As far as I know, Harvard still does not use the term "major" at all, and Teddy would not have had one.

The second half of the nineteenth century was a time of vast change in the educational system in the United States. Even schools that we today view as world-class research universities (such as Harvard, Yale, etc.) were until the middle of the nineteenth century almost entirely concerned with educating future generations of clergy. In Harvard's case, that clergy was the Unitarian clergy, the dominant strain of Christianity among the social and financial elite of Boston, and a direct descendant of Puritanism, though theologically quite distinct. The great American thinker Ralph Waldo Emerson, for example, was trained as a Unitarian minister at Harvard.

The first crop of Americans began to receive PhDs and other professional academic degrees in Europe (mostly Germany) in the time just before the Civil War, and they came back to the U.S. and installed themselves at American Universities, beginning the process of changing those institutions into a form that we would more readily recognize today. But in the 1870s Harvard was just beginning to throw off its history as a religious institution - for example, mandatory chapel was not removed as a requirement until after T.R. graduated. It would still have been quite small, all male, and all white. Most of the students would have been scions of prominent Boston families, and many of them were destined for careers as Unitarian ministers.

Sources:

Here are a couple of Harvard-published sources that can give you a general sense of how the university evolved:

http://www.harvard.edu/about-harvard/harvard-glance/history/historical-facts

http://www.harvard.edu/about-harvard/harvard-glance/history-presidency/charles-william-eliot

There are a number of sources on the development of American higher education. For the late nineteenth century I like Diehl's The Americans and German Scholarship, even though it is getting a bit long in the tooth.

(Edited for grammar).

rip I'll never find out

Checked out some books on TR from my local library, wish me luck.