You will receive an email with a link. The link takes you to a page that prompts you to pay before going any further. The application fee is $38, which will go toward funding the mission.
The next part is slightly more labor intensive. It required making a short (30-70 second) video. In the video, you must explain why you want to go to Mars and why you make the perfect candidate.
The next part asks for a lot personal information. For now, I revealed the bare minimum in order to move onto the next step. I might spill more details if I move onto the next round.
The private questionnaire asks a series of college-application-like questions. Since it's a one-way trip, they want to know how you'll deal with leaving your family and friends behind. Obviously you should be someone who adapts well.
Then, submit! There was no indication of when I will hear back. In the meantime, you can check out the competition over on the Mars One homepage, which features the application videos of future Mars travelers.