I applied online. The process took 5 days. I interviewed at SpaceX
Interview
Contacted via phone and e-mail on Monday evening. I had other offers pending - HR department was incredible and set-up a phone interview the following day (Tuesday). I waited a single day and heard back on Wednesday night that I was selected for an internship. After submitting pre-offer forms, I was sent on offer on Friday night and accepted. SpaceX's recruiting team is impressive given the sheer amount of people they are dealing with.
The communication in-between was lacking, but totally understandable: they are dealing with tens of thousands of applicants. The majority of the e-mails and phone calls were late evenings (Pacific time).
The phone interview lasted 30 minutes and was focused on my previous/current internships, a little bit on my previous coursework, and my background in other jobs (I am an "old" student). The interview was fairly casual and free-flowing. I had many, many questions about SpaceX's materials department and asked them during and after the interview.
The phone interview went well enough that a second interview was not necessary (the other offers also created a time crunch, which probably also impacted their decision).
Overall, the interview was "easy" but mainly because I demonstrated an extreme interest in the company and... this my dream position. Come prepared with a great deal of curiosity on what SpaceX is doing now and in the future - show you are prepared and interested in helping them advance their vision. There are countless intelligent and hard-working engineering students with the picture perfect resume - show that you are the special case that is committed to helping SpaceX get people to Mars, permanently.