There seem to be a lot of different levels of preoccupation, routines and rituals. Depending on each person it always means something else to them as individual. I'm not sure how to answer your question exactly, because I wonder which part it is that confuses you. So my answer is a bit of a guess.
I was diagnosed with Aspergers. My preoccupation: Once I liked or was fascinated by something it was very hard not to constantly think about it, analyze, find out more... Letting go of it was hard, often not wanted and doing homework or what ever had to be done, because it was expected took a lot of focusing.
Inflexible adherence - not sure exactly, but for example if I wanted to understand something I would keep asking until it was explained in a way that fitted with my way of understanding. As parents will know this can take an extremely long time.
Nonfunctional routines or rituals: Love routines, they are predictable - nothing better - cannot say I ever had a nonfunctional, because if it wouldn't work for me I wouldn't do it, right? Rituals are fine too. It always feels right to know what's going to happen.
Keep in mind it's different for everybody and from what I read so far these things can be anything.
Yes, that's exactly the kind of reply I was hoping for - thanks Kasi!
Please, everybody else - feel free to chime in. I need more examples in order to understand these criteria and recognize how they apply to me.
I am not an Aspie, but I can tell you what I go through with my friend who we believe has Asperger's Syndrome. Some examples of behaviour with him are, he takes alot of things literally which makes it very difficult not to have misunderstandings that can often lead to arguments until it's explained to him several times. My Aspie friend also gets hyperfocused on what interests him which seem to be two or three things only and he will talk about those things for hours and sometimes weeks/months not giving the listener consideration or letting anyone chime in and give their thoughts. It is almost like we have to force our point of veiw. Obsessive compulsive behaviour is a big one with my friend, very interested in measurements and patterns and very poor coping skills concerning stressful events. There are so many other things but these are a few of the most prominant behavioural issues that we have noticed.
I should add something, my Aspie friend is a wonderful person and has a good heart. These behaviours are not done maliciously that anyone can tell, but more that he doesnt realize what he is doing to those around him due to his trouble understanding emotions of others unless he has felt it himself. Not to say that he has no empathy at all but sometimes it seems that he does not have very much.