I found out about the internship through a posting on my university's career center. I emailed ther person directly responsible for hiring with my cover letter, resume, and writing sample. A couple of days later, I was invited in for an interview.
The interview began with at 1:1 interview. For the most part, it was a typical behavioral interview with questions like: Describe a time when you took a leadership role in a project; Describe a time when you had to deal with a bad superior; etc. I was also asked why I wanted the position and how it fit in with my studies. Afterward, another employee came in and asked me questions about my academic background. In total, the interview lasted about an hour.
A couple of days later, I was invited to interview with the team's vice president. It took about 30 minutes and the questions were very similar. A day later, I had an offer.