An optical vortex is a singularity in a wave of light. There are both experimental and theoretical concerns about optical vortices.
Experimentally, optical vortices can be produced simple by generating pure Laguerre-Gaussian transverse laser modes. These modes, in their pure form, have an azimuthal phase term, which gives them a phase shift around their circumference of 2*pi*l, where l is one of the parameters of the mode (the other, p, is a measure of the radial minima).
But since the shift around the circumference is constant, looking closer to the center will yield faster changing of the phase. This is a spiral, or vortex (hence the term optical vortex - after all, this is light). At the center is a singularity.
Theoretically, there is no consensus as to the physical meaning of this singularity. For a quick, visual insight into the problem, just look at M. C. Escher's "Print Gallery". Escher had no idea what to put in the center. Analogously, theorists have no idea what to put in the center of their LG beams. And since no experiment can either perfectly realize a pure LG mode, we can't just look at one of these things. Incidentally, a recent New York Times Science Times article explains that a mathematician has solved the problem of the center of "Print Gallery".