Secular people believe we are physical beings, and that things like thoughts, or love, for examples, are simply a series of chemical reactions. Proponents of the tripartite view claim that there is a clear difference between soul and spirit, though they may be so intertwined and similar that they would be hard to separate without scriptural clarity. Opponents argue that there is no real separation here (though there must be some difference, at least in emphasis, if two different words are used), but the two are only used as a metaphor of things hard to differentiate, like the thoughts and intentions of the heart. Personally I think we have three parts. In any case the terms are useful in thinking and discussing us.
The body is physical - although the complexity of the microscopic "circuitry" continues to amaze with things we cannot see with our eyes. I wonder what the evolutionist from the middle of last century would say - he said he'd have to give up on evolution if they ever found a wheel in nature.
The soul includes the mind, emotions, will and so on. While the body is amazing, the soul almost goes beyond our ability to even talk about it's most incredible features.
Finally there's the human spirit. Obviously if the soul is difficult to talk about, what can we do with understanding this? It is very difficult to get our minds around. That's not really surprising given language is a tool of the soul. Far greater minds than mine (e.g. CS Lewis in his sci-fi trilogy) have attempted to describe at least the feel of some aspects of it - but the best of words cannot do it justice.
One of the things things that intrigues me was referred to in the quote from Wikipedia included in the first paragraph. I'm just starting to learn of the interaction between these three parts. It's well known that mind and body, for example, are connected in ways we only vaguely appreciate. When we are hurt, it affects our outlook. When we're depressed (which can have causes in one, two or three parts of our being), it affects our bodies.
When I had a stroke, it obviously affected both body and mind. But it was only when I had clean blood (after my wife gave me a kidney) that I realised I had no sense of spiritual reality at all. I still don't understand this, but six months after the transplant, my first time into a meeting with Christian music playing, I "felt" God's presence as I used to. Not only that but during that weekend I learned concepts I'd never noticed before, and started to discuss things with God in a way I'd never known previously.
I'm still figuring this out - it's more complex and hard to understand that I ever appreciated. Both the intertwined parts and the capacity of the individual parts are far more than I have appreciated so far - and I look forward to finding out more about them as I move forward.