Most graduate school applications for American graduate programs are due in December or January for a September start.
Some types of programs, especially law school, are rolling applications, meaning that the earlier you apply, the earlier you’ll hear back. In this case, you should aim to apply as early as possible to maximize your chances.
Medical school follows a completely separate timeline with much earlier deadlines. If you’re applying for medical school, you should speak to advisors at your university for more information.