A "bicycle college" (also school, academy, etc.) was an institution devoted to imparting the basic skills of cycling, as today an individual might attend driver's education classes before getting behind the wheel for the first time; they also often featured a rink (closed circular circuit) for safe instruction, racing, and general entertainment, with bicycles for rent. David V. Herlihy notes in Bicycle: The History discusses rinks in particular (113):

By spring 1869, the largest cities hosted a dozen or more of these "academies." Most mid-sized cities had at least two. Even small towns were rarely deprived of a facility. Eau Claire, Wisconsin, had but one rink with two bicycles, yet it still managed to stage an exciting race. Some of the larger rinks had as many as fifty machines, including the top New York models. But a typical fleet comprised about a dozen machines.

Bicycles (or velocipedes) were already big business in France and growing in the United Kingdom, and by the late 1860s were starting to catch on in the United States as well, through the introduction of clubs, athletic events at Ivy League schools, and improvements to technology - we take a lot of the technology of bicycle as a given today, with its rubber tires and hand-mounted brakes, but in the 19th century the form of bikes was far from standardized, the machines often heavy, expensive, and requiring no little skill from an adult rider learning how to use one for the first time. Frances Elizabeth Willard in A Wheel Within a Wheel: How I Learned to Ride the Bicycle, with Some Reflections By the Way (1895) discusses her efforts to learn to ride by the teachers she had:

I studied my various kind teachers with much care. One was so helpful that but for my protest she would fairly have carried me in her arms, and the bicycle to boot, the whole distance. This was because she had not a scintilla of knowledge concerning the machine, and she did not wish me to come to grief through any lack on her part. Another was too timorous; the very twitter of her face, swiftly communicated to her arm and imparted to the quaking cross-bar, convulsed me with an inward fear; therefore, for her sake and mine, I speedily counted her out from the faculty in my bicycle college.

Willard here is not referring to a specific institution or course of formal instruction, but simply the whole practice of learning to ride a bicycle.

As far the exact institution referred to as opening on December 5th, 1868 in New York City, I believe that Herlihy refers to it (105):

In the fall of 1868, a furor erupted in Manhattan. A local athletic club hosted an inaugural bicycle contest in November, and specimens began to appear in Central Park, to the astonishment of strollers. In December, the Pearsall brothers opened a riding school on Broadway that immediately attracted several hundred prominent citizens, including doctors, lawyers, merchants, and ladies.

This would be, according to a contemporary advertisement reproduced in the book (116), the "Grand Velocipede Academy" or "Gymnacyclidium," and declared itself the first such institution in America.