I interviewed with the APA a few weeks ago. It was a relatively easy interview process. First, I met with the HR Director; I actually shown up about 15 min. early for the interview and she had me waiting in the lobby for another 40 minutes! She was extremely nice though; she just looked over my resume and took down a few notes while making small talk. She then told me that I was to interview with my would-be supervisor. So, I waited another few minutes until my interviewer came to get me and take me to her floor. She was very nice and funny; although she read off a prepared sheet of questions, it was a very easy and relaxed interview. She asked me the usual questions: What were your responsibilities at your previous job? Why should I hire you over any other applicant?
Just be prepared to answer questions about your job capabilities rather than so much about the company. However, you should read up on the specific office that you are interviewing for (i.e. education, women, accreditation, etc.). Each office has its own specific webpage seperate from apa.org.
After interviewing with my would-be supervisor, she took me over to her boss' office for the final stage of the interview. Her boss was very sweet and asked me just a few questions, one of them being "What would you do if you were in a situation at work in which a colleague had a differing opinion regarding how to handle a specific project?"
After the interview, my would-be supervisor asked me to write her an e-mail explaining why I wanted the job. She was very straight-forward and said it was to measure my follow-up and writing skills. About 10 days (which was a little longer than I should have waited) after sending the first e-mail I sent a second. I got a response right away saying that they were still in the interviewing process and I should expect a call soon. I know that they have called up at least one of my references, so I am hoping for a job offer this week.