Rational, nonrational and irrational

What does it mean to think rational, nonrational or irrational? To speak about this I have to accept that we can divide the way people think into this three category, and no more. First, I have to define those words as simple as I can, but not too simple, that will not help. Second, I have to compare them to see the similarities and the differences, and only after this can I prove the uselessness of this separation.
What is rational? I would use the Ed. Miller's explanation, that rational thinking "[...] seeks to eradicate [...] ignorance, superstition, prejudice, blind acceptence of ideas, and any other form of irrationality. It presses us to coherent and valid expressions of our ideas." I would say that rational thinking is based on examination, explonations and reasons. A rational thinker is someone who first waits and sees what happens, then go step by step to expain what had happened and why did that happen, always connecting an idea to another, making his reasoning coherent...