I was initially contacted by a recruiter via LinkedIn. After an initial phone call to discuss my background and interests, went through 2 phone interviews. These include a detailed discussion on my experience and specifics on projects I worked on (be ready to explain what you did, how, and why you did it), as well as some coding. Amazon seems very interested in understanding the complexity of your work and how versed you are in sharing those experiences.
After these interviews, I was scheduled for an onsite interview at the Seattle HQ. Amazon puts you in contact with a travel desk that arranges your flight and accommodations for you. Feel confident in asking for flights/airlines or other requests you may have, as they are friendly and will try to get what you ask for. Amazon flies you a day before the interview and puts you on a very nice hotel 10 min away from the office.
Onsite interview day included five 1:1 interviews, including two development managers and three SDEs, focusing on different areas. You are received by your recruiter and have a 30 min chat with him. You get some details on what to expect during the day, and the days after the interview.
Most interviewers will be writing notes while you chat (they ask if you're ok with it - I wonder if you say no, what happens?!). Questions include behavioral and technical, as well as coding on a whiteboard. Interviewers are friendly and help you through the process - ask for what are you thinking and give you hints in case they see you getting stuck. The best way to easy the nerves is to keep talking as you tackle problems. It helps more than what you think.
I heard back from my recruiter after 2 days saying the team wanted to move forward and extend an offer. A couple days after I got the details of it and after some light negotiation I accepted.
My advise is to avoid focusing too much on the horror stories you find online, and just make the experience your own. It is unlikely you'll get something like you have read here. It will be unique and a good learning experience, whatever the outcome is. Just try to enjoy it as much as you can and please, don't go into it without serious preparation before hand. Even if you don't make it you want to give a good impression, and of course, avoid post-interview "what if I have..." thoughts.