I find the Myers-Briggs and related personality-type constructs to be fascinating. Of course, we don't want to stick people into rigid categories that may not capture all the nuances of who they really are, and we don't want to bind our lives to self-fulfilling prophecies based on what sort of personality-type we think we are. With all proper warnings and nuances in place, however, it seems to me based on observation of myself and others that this personality-type sort of thinking is indeed capturing a real component of how human personalities work. The tests I've seen (and I've never paid for one) seem to be fairly accurate at placing people into various types, and sometimes they can be astonishingly accurate at painting a picture of what a person is like.
Probably the most amazing example of this I have seen so far relates to myself. The description of the INTP personality-type seems to capture core aspects of my personality with astounding precision. If you want to understand me, read this article and this article describing the INTP personality. Of course, even if we fit into a particular personality-type fairly well, we are all unique, and we have different beliefs, values, and experiences and can make different choices that affect how we think and live and interact with others. So no personality-type portrait is going to capture or accurately describe all that we are. But it can often capture a good chunk! That is certainly the case with me and INTP!
If you want to take a free test that seems to work pretty well from what I've seen, go here. (I will say that the test seems to slant a bit toward the J side; so if you find yourself labeled a J, you might read the corresponding P description as well and see which one fits better.) If you do this, let me know how it went and what you found!
My daughter Erin, 13, just took the test today, and it labeled her an ISTJ. It does seem to fit, but she is a little young for the test, so perhaps she can take it again in a few years and see if it still does.
UPDATE 7/1/13: I thought I'd mention also that my wife, Desiree, seems to be an INFP.
UPDATE 5/5/14: Another good description of the INTP personality.
UPDATE 12/4/14: And another good description. And this is interesting too.