I've heard the work a few times now, the most recent in the star trek film. Is a singularity a real thing? If so what is it?
The word "singularity" is generally used to denote that some quantity becomes infinite or in general becomes undefined (i.e. cannot be expressed as a finite real number).
One particularly common use is in general relativity. At the event horizon of a black hole (the surface of no return), often coordinates that were well behaved far away run into trouble. For example, the time as measured by a clock infinitely far away slows down and comes to a halt near that surface, so far-away observers see stuff falling into a black hole slower and slower, never quite making it.
However, the above is only what is known as a coordinate singularity - something went wrong with your coordinate system, but nothing physically "broke." For a more benign example, the North Pole of the Earth has a coordinate singularity if you use standard longitude and latitude. What is the longitude of the North Pole? Well, it's undefined. There is nothing physically important about this statement - it merely reflects the choice of a poor coordinate system for that region. (In fact it can be shown mathematically that no sphere can be covered entirely by a singularity-free coordinate system.)
There can also be "true" singularities. In general relativity you might compute some coordinate-independent quantity - the (scalar) curvature of spacetime for instance - and see that it goes to infinity. No change of coordinates will "fix" this, and it means physically meaningful quantities are behaving badly. In classical general relativity, it has been proposed (see the cosmic censorship hypothesis) that any such true singularities (as might exist inside the event horizons of black holes) are always shielded by event horizons, so information about them can never reach us. In models where this is true, it doesn't even make much sense to talk about the existence of such things, since their properties by definition can never be studied.
One can also cast singularities in terms of geodesics - the paths that objects, light, and anything else follow in spacetime when no other forces act on them. A singularity is a point where a geodesic just ends. You reach that point and then that's it - the laws of physics have nothing more to say about your position.
In these latter two cases, if the singularities are "real" and somehow not hidden from us, they signify either (1) the universe has stopped being well defined, and therefore all bets are off when it comes to predictability and science in general, or (2) our theory is incomplete and needs more work. Most people believe in choice (2).