Sympatric speciation is simply speciation that occurs when a species splits into two reproductively isolated groups without any physical barrier arising between those groups. It's often treated as the opposite of allopatric speciation, in which a species is split by some external barrier (a new mountain range, a river, &c) and the separated populations evolve on different trajectories until they're unable to exchange genes even if the barrier is removed. There are a number of ways biologists think this can happen - for instance, a population of insects using two different, co-occurring host plants, might split into two species on the different hosts - but not many good cases where we're pretty sure that it has happened.
In the new article, Kirkpatrick and his coauthors argue that the problem is one of definition. Sympatic speciation, as a concept, is set up to be impossible to test conclusively: Although it's easy to show that two closely-related species occupy the same territory in the present, it's rarely possible to show that they became different species while they were occupying the same territory at some point in the past, much less that they were "panmictic," or freely interbreeding.
The problem for empiricists is that biogeographical sympatry is relatively straightforward to diagnose, but the initial condition of panmixia specified by population genetic models is virtually impossible to test.As an alternative, the authors suggest ignoring the Platonic ideals of "allopatric" versus "sympatric" speciation, and instead concentrating on the interaction between divergent natural selection and gene flow in causing or preventing speciation. Not only is this sensible, it's what most evolutionary ecologists are doing right now, anyway.
B.M. Fitzpatrick, J.A. Fordyce, S. Gavrilets (2008). What, if anything, is sympatric speciation? Journal of Evolutionary Biology, 1452-9 DOI: 10.1111/j.1420-9101.2008.01611.x