I am an INTP:
Introverted 22%
Intuitive 62%
Thinking 12%
Percieving 33%
INTPs are pensive, analytical folks. They may venture so deeply into thought as to seem detached, and often actually are oblivious to the world around them.
Precise about their descriptions, INTPs will often correct others (or be sorely tempted to) if the shade of meaning is a bit off. While annoying to the less concise, this fine discrimination ability gives INTPs so inclined a natural advantage as, for example, grammarians and linguists.
INTPs are relatively easy-going and amenable to most anything until their principles are violated, about which they may become outspoken and inflexible. They prefer to return, however, to a reserved albeit benign ambiance, not wishing to make spectacles of themselves.
A major concern for INTPs is the haunting sense of impending failure. They spend considerable time second-guessing themselves. The open-endedness (from Perceiving) conjoined with the need for competence (NT) is expressed in a sense that one's conclusion may well be met by an equally plausible alternative solution, and that, after all, one may very well have overlooked some critical bit of data. An INTP arguing a point may very well be trying to convince himself as much as his opposition. In this way INTPs are markedly different from INTJs, who are much more confident in their competence and willing to act on their convictions.
Mathematics is a system where many INTPs love to play, similarly languages, computer systems--potentially any complex system. INTPs thrive on systems. Understanding, exploring, mastering, and manipulating systems can overtake the INTP's conscious thought. This fascination for logical wholes and their inner workings is often expressed in a detachment from the environment, a concentration where time is forgotten and extraneous stimuli are held at bay. Accomplishing a task or goal with this knowledge is secondary.
INTPs and Logic -- One of the tipoffs that a person is an INTP is her obsession with logical correctness. Errors are not often due to poor logic -- apparent faux pas in reasoning are usually a result of overlooking details or of incorrect context.
Games NTs seem to especially enjoy include Risk, Bridge, Stratego, Chess, Go, and word games of all sorts. (I have an ENTP friend that loves Boggle and its variations. We've been known to sit in public places and pick a word off a menu or mayonnaise jar to see who can make the most words from its letters on a napkin in two minutes.) The INTP mailing list has enjoyed a round of Metaphore, virtual volleyball, and a few 'finish the series' brain teasers.
INTPs in the main are not clannish. The INTP mailing list, with a readership now in triple figures, was in its incipience fraught with all the difficulties of the Panama canal: we had trouble deciding on:
1) whether or not there should be such a group, 2) exactly what such a group should be called, and 3) which of us would have to take the responsibility for organization and maintenance of the aforesaid group/club/whatever.