You might think that just getting an initial interview is a good sign that you are going to get the job you want; but really, it's only the first step. Don't get me wrong, if you are even chosen for an initial interview, it means the employer thinks you have some potential to succeed at the position you have applied for. But really, the first interview is really just, as harsh as it might sound, a way to weed out the merely good candidates from the ones that are truly exceptional. Basically, it's a screening process.
I've already described what you should expect from your first nonprofit job interview, so what exactly should you expect from the second (or the third)? In general, the second interview at a non profit is where the employer will begin to question you for specific instances to back up what you have said in your interview. As long as you didn't actually lie in your resume, then this shouldn't be a problem for you. Basically, you are going to be asked to recount experiences at your previous job where you made a difference for the company. Once again, it is important to be honest here. The interviewer is only going to be asking you things that were apparent from your resume. If you hesitate even a little bit, it will seem like your resume was not entirely honest.
The second interview is also a time to get you more acquainted with the specific responsibilities that the position will entail. This would also include the type of compensation you would receive. It is important to make sure to have your interviewer elaborate on any of the things they explain if you don't understand it. Remember, as the old cliche goes, the only stupid question is the one not asked.
Non profit organizations will have no more than three interviews. If a third interview does happen, it will be the final interview. This interview will be to finalize the employer's views on you as a person and employee. If you get to a third interview, you are well on your way to getting the job. All you need to do at this point is be confident, and show that you are ready for any challenge the company will throw at you. Whatever you did on the second interview must have impressed them so really, don't change anything you have done. It all boils down to this: be yourself.