I remember taking the Regents when I was in HS (in the 1980's). The exams seem to be pretty good. They've been around for decades and change very little over the years.
There are plenty of review guides available and it is very clear to teachers and students which material and types of questions they need to master to do well on the test.
Info and Sample tests with answers here:
New York Regents Exams
NY Regents (Integrated Algebra)
NY Regents (Geometry)
NY Regents (Algebra II/Trig)
NY Regents (Mathematics B)
New Jersey used these for the past two years and ended up not counting them.
ADP Algebra I End-of-Course Exam
ADP Algebra II End-of-Course Exam