Astronomy graduate programs provide funding for their students through stipends. The stipend package will vary from school to school, and will be given to you in detail when you are admitted. The most common types of funding are Research Assistantships (RA) and Teaching Assistantships (TA). Students may also be offered university or departmental fellowships.
NSF Graduate Research Fellowship Program:
The National Science Foundation provides support for graduate students who are selected during their fellowship application process. These fellowships provide support for 5 years, three of which come with $34,000 of funding per year. The NSF has recently changed their application policy such that students are eligible to apply once in their senior year of undergrad and once during graduate school. You should strongly consider applying during the Fall semester of your senior year. Applications are usually due at the end of October. The application requires a personal statement, research statement, general information about your education, and a transcript. For more information about applying, please see nsfgrfp.org.
A list of Major US Fellowships for Astronomy Students from Astrobites