The application and interview processes is a little different in every Congressional office. However, you should expect to submit a resume and cover letter.
Your Resume: Offices will not pay attention to your application if your resume is sloppy. Make sure it is formatted well and is free of typos. The formatting does not need to be fancy, but it should be obvious that you put effort into it. I modeled my own resume after the example here.
Your Cover Letter: This is your chance to shine. Talk about why you want to be an intern, your passion for politics, and your admiration for the office you are applying to. Make sure your personality and passion shine through your cover letter. Cover letters all follow a standard format.
I have included the resume and cover letter I used to apply to my first hillternship below. They are not perfect, but they got me the job.