Regents exams are the standardized testing given to high school students in specific subjects starting in freshman year. A regents diploma (meaning you passed the required exams) is a higher level diploma then a standard high school diploma. The regents diploma with honors is for students who pass additional tests before graduation. An 85 or better (at least when I was in school) was, from what I was told, was considered mastery in the subject matter. I believe Texas, Florida, California, and Massachusetts all have state requirements too. The general diploma at my school was seen as equivalent to a GED.
Barrons (exam prep company) has study guides we often used to prepare for the exams. The standardized testing makes a NY education (in good school districts) seen as a more impactful and rigorous education compared to lower ranked education systems in other states.