There are three main types of school options for kindergarten in NYC -- public, charter and private. (If you’re instead interested in homeschooling, you can find more information about that here.)
Public schools are publicly funded and operated -- applications are accepted/ranked all at once at myschools.nyc. Charter schools are publicly funded, but privately operated -- their admissions processes are specified on their school websites, but there is also a common application on the New York City Charter School Center website. Private schools are privately funded and operated. Visit each private school's website for more information about their application process.
Data source: NYC IBO, 2019 Report
As you can see, the vast majority of K-12 students attend public school, but there are varying benefits that come with all three options. To help you understand if one (or more) of these school types makes sense for your family, please find a side-by-side comparison below with some of the key factors that you may want to consider.
Lastly, it's important to point out that there's a small distinction between private and independent schools, though they're often referred to synonymously. I try to do my best to use them each appropriately throughout this website, but I may slip from time to time. My apologies in advance!
In the meantime, here's some information so that you can do a better job than I when referencing them: