What is a sudden singularity? I've seen references to some sort of black hole (or something) referred to as a sudden singularity, but I haven't seen a short clear definition of what this is for the layman. 
 A: A sudden singularity is a singularity that forms in the universe in a finite time.
This may see like a strange definition. After all, don't the singularities in black holes form in a finite time, so shouldn't they, and indeed all singularities be sudden? Actually, no! For observers like us, floating around the universe, in our frame of reference it takes an infinite time for a black hole to form.
This apparently paradoxical result has been too widely discussed, in this site and elsewhere, for it to be worth me elaborating on it here. An observer falling into the black hole will encounter the singularity in a finite time, but for observers outside the event horizon the singularity never forms. The type of big rip singularity usually referred to as a sudden singularity forms in a finite time (for all observers) because we are in effect falling towards it just like the observer falling into a black hole.
A quick footnote:
As Ben points out in his answer, my definition above is too broad because it includes singularities (like the Big Crunch) that are not sudden. as described in Nojiri's paper, sudden singularities are just one of the four types of singularities that can occur at non-zero scale factors (i.e. not the Big Bang or Big Crunch). I'm not sure how usefully this can be summarised for the lay reader since it's all a bit technical. @Skotch, I would have a look at Nojiri's paper and maybe ask a further question if you want anything in it carified.
A: People have known for a long time about the existence of cosmological models that include a Big Crunch singularity. In these models, the density of matter in the universe is great enough to cause it to recontract. These models are no longer of interest as descriptions of the actual universe, since we now know that the expansion of the universe is accelerating rather than decelerating, presumably due to dark energy with poorly known properties.
In fact, if dark energy turns out to have certain characteristics, then it's possible that the expansion will blow up at some point, so that all matter is destroyed by being torn apart. The most commonly discussed scenario of this type is called a Big Rip, but there are other scenarios as well, classified in Cotsakis 2004. A singularity like this has two properties: (1) it happens at a finite time in the future, and (2) it's a blow-up, not a recollapse like the Big Crunch. There can be cases where the cosmological scale factor $a$ blows up, and others where $a$ stays finite, but $\dot{a}$ blows up. For comparison, a Big Crunch would have $a$ going to zero.
The term "sudden singularity" seems to be used in the literature for one very specific type of blow-up. Cotsakis says, "For instance, they may correspond to ‘sudden’ singularities (see [19] for this terminology)," where reference 19 is Barrow 2004. Barrow constructs an example that is not driven by dark energy at all. The strong energy condition holds, and both the density and the scale factor are always constant. However, the pressure and curvature blow up to infinity at a finite time. He gets this behavior by not requiring any equation of state to hold between the pressure and the density. Since both Barrow and Cotsakis seem to be defining the term based on one example (the word "sudden" only appears in Barrow's title), it's a little hard to tell exactly how broad they intend this definition to be. Nojiri 2005 defines a sudden singularity as one in which the pressure blows up to plus or minus infinity at some finite time, while the scale factor and density stay finite. This implies that a sudden singularity can never occur if there is a definite equation of state.
John D. Barrow, 2004, "Sudden Future Singularities," http://arxiv.org/abs/gr-qc/0403084
Spiros Cotsakis, Ifigeneia Klaoudatou, 2004, "Future Singularities of Isotropic Cosmologies," http://arxiv.org/abs/gr-qc/0409022
Nojiri et al., 2005, "Properties of singularities in (phantom) dark energy universe," http://arxiv.org/abs/hep-th/0501025
