Currently, Rupert Giles is the head director of the Watcher's Council and therefore the school in NYC. After the Council were attacked, Giles went to work rebuilding and eventually successfully revived the Council, establishing several schools/bases in all major cities. The WCTA of NYC was created, stabilized and established before Buffy and the gang decided to relocate to NYC. Giles more closely concerns himself with the affairs of this branch of the school now that he lives in the city.
In NYC, the school takes place in the large hotel and surrounding grounds of an English tycoon who moved to NYC upon request of Giles. He had no problem buying up the old and no longer used building and turning it into a school - he had the money and resources. Like all other watcher's schools, the hotel hosts an entire database of supernatural information. Below the hotel itself is a large headquarters, much like the Slayer Organization - with state of the art training equipment, large library, lecture halls and classrooms. The tycoon currently runs operations of the school. Like before the Great Revelation , the school remains secret.
Potential Watchers are "called" by the Council and required to attend Watcher's Academy. They are required to study only "until they're ready" and not for a set term. Watchers are trained for years in supernatural phenomena, lore, and legend. They are taught the ends and outs of protection from supernatural entities as well as extensive witchcraft and combat skills. Watchers live in the hotels rooms and attend classes literally all day.
Previously, schools used to be a stiff watcher's bootcamp - a nightmare consisting of years of constant studying and training with little to no rest, uncomparable expectations and grading procedures, and a cut-throat failing rate. Not even the most prestigous ivy league universities could compare with the watcher's council - stress level wise or academically. In addition, watchers were taught to not be 'parents' or friends to their slayers. They must remain detached and perform demeaning and crippling tests on them to be sure they're up to par.
But now that Giles is in control of the council, those policies are no more. Schools are more reasonable, with a steady-paced curriculum and high expectations, yes, but it is much less cut throat. Tthe work and study is still challenging, but the school's atmosphere now actually feels like a school, as opposed to an academic military.
Roger Wyndum Pryce - father of former watcher Wesley, was the only one who survived the attack on the Council that destroyed them and has made it inescapably known that he does not agree with current operations.