The standardized tests here are called the New York State Regents Exams, taken in 4th grade, 8th grade, and throughout high school. I am not quite sure if "NYSRE" is what it's better known as, though.
In 4th grade, tests on English and Math are taken in January and June, respectively. Parents have spoken against these, as children who don't pass them have a good chance of being held back a year. This actually happened to a friend of mine who is 2 months older than me but is only a junior in high school.
In 8th grade, tests on the same subjects are taken, but at a higher level, of course. I've not heard of any parents having problems with these, though.
In high school (pretty much every grade except 12th, as most seniors have either hardly any classes or AP classes which have AP exams instead), there are tests in Math, Science, English, LOTE (Languages Other Than English) and Global/National History, depending on what courses you're taking and at what level.
The only exam freshmen have to take is a proficiency exam in whatever language they chose.
Sophomores and juniors get the most exams.