A
singularity that is not surrounded by an
event horizon. For example, consider a
black hole with an electrical
charge (a "
Reissner-Nordström black hole"). For charges less than a certain threshold, the hole has two event horizons, one inside of the other. For a charge equal to the threshold, the radii of the two event horizons are equal, and for a charge greater than the threshold, the hole has no event horizon at all, a naked singularity.
The possibility of naked singularities is troubling because although singularities themselves are troubling the problems with them have always been swept under the rug by saying that they are always "walled off" from the rest of the universe by an event horizon. That means that nothing that happens near a singularity could ever effect us out here in the rest of the universe. The singularity is unobservable, and hence unimportant. The possibility of naked singularities changes all of that, since one could, at the very least, observe radiation from the vicinity of the singularity. Suddenly singularities do matter, and the debate over whether they are "real" or not takes on new relevance.
There is intense debate over whether naked singularities can actually form in practice, and at one point two famous physicists, Kip Thorne and Stephen Hawking, actually had a bet going on the subject. The Reissner-Nordström black hole is kind of a bogus example, since it's hard to imagine how a black hole could accumulate that much charge, and it seems likely that it would quickly attract enough opposite charge to cancel its initial charge. Numerical calculations have shown another way that a naked singularity could arise, involving gravitational waves converging on a single point, and that was evidently enough for Hawking to concede the bet, although he maintained at the time that the situation was pathological. Consequently, it is still not certain whether naked singularities can actually form or not.