I believe they talk about the computational task described in [1] (published in 2016). Note that this is somewhat contrived: it's a task defined specifically so that it is very difficult for classical computers and relatively easy for quantum computers. It is also defined in a way that ensures that a quantum computer necessary to demonstrate quantum supremacy is relatively small in terms of the number of qubits.

The task is that of generating output samples from pseudo-random quantum circuits. The paper shows that the task has exponential computational complexity on classical computers and that it can be used to demonstrate quantum supremacy using a quantum computer with approximately 50 qubits.

Now, the IEEE Spectrum article indicates that the team has now built a 49-qubit quantum computer and plan to use it to demonstrate quantum supremacy.

EDIT: Replaced direct pdf link with the article page link.

[1] https://arxiv.org/abs/1608.00263

I don't know a good link to a discussion of this system, but the idea of establishing quantum supremacy has been much discussed in optical systems under the title of boson sampling.

https://en.wikipedia.org/wiki/Boson_sampling

It wasn't clear to me either. Demonstrated "supremacy" over conventional general-purpose CPUs, as described in the article, does certainly read like hyperbole.

Here's an educational video on the subject that explains it in a way that everyone can understand. https://www.youtube.com/watch?v=JhHMJCUmq28

Here is another link that, while being a tiny bit mathematically rigourous, made me have an "a ha" moment about how quantum computtattion works.

PBS Digital Studios is truly a treasure and really shows what's possible with eduxation when you truly take advantage of internet video.