Neutrons come out of a fission event at high speed, about 20% the speed of light. These are called fast neutrons. In almost all nuclear reactors we've built, we deliberately slow these neutrons down with a moderator material (which must have light atoms in it: things like graphite, water, beryllium). It's much easier to get a pile of nuclear fuel to chain react when you slow neutrons down, largely because Uranium-235's liklihood of absorbing a slow neutron is much higher than its liklihood of absorbing a fast neutron.

(Aside: slow neutrons are often called "thermal" neutrons because they're in thermal equilibrium with the atoms around them. At room temperature they're going 2200 m/s, which corresponds to an energy of about 0.0253 electron-volts, slowed down from 2 million electron-volts when they emerged from fission)

A fast-neutron reactor ("fast reactor") is a reactor where the neutrons that emerge from a fission event are kept moving fast, simply by not putting a moderator into the design. These reactors use heavy nuclei or very low density material as coolants (sodium metal, lead-bismuth eutectic, helium, etc), so as to not slow down any neutrons. It takes much more fissile fuel to get a pile of nuclear fuel chain reacting in this configuration, but once you get it going, you get some very nice benefits.

Once started, fast neutron chain reactions have a vast surplus of neutrons going around. This is for two main reasons: (1) The number of neutrons emitted per fission increases dramatically as the neutron speed increases above a threshold around 1 MeV, and (2) parasitic absorption of fast neutrons is low for all nuclides. With so many extra neutrons around, you can afford to put extra "stuff" in your reactor, such as spent nuclear fuel (nuclear waste) or extra fertile material (uranium-238 or thorium-232).

You can make these reactors into breeder reactors (which convert U-238 or Th-232 into fissile fuel in such a way that could power the entire of humanity 10x over for a few million years, at least, using known resources), or you can make these reactors into burner reactors, whose job it is to transmute used fuel from other reactors into shorter-lived material (similar to what OP's article is about, but much more practical).

A definitive guide to understanding transmutation of spent fuel in fast-neutron reactors is put out by the UN's International Atomic Energy Commission [TRS-435]:

"Implications of Partitioning and Transmutation in Radioactive Waste Management"
https://www-pub.iaea.org/MTCD/Publications/PDF/TRS435_web.pd...

A handful of fast-neutron reactors have been built, starting with tiny single critical mass assemblies in the 1940s, to the first true demo of breeding more fuel than you consume in EBR-1 in 1951 to the EBR-2's safety demo weeks before Chernobyl showing that the reactor could passively shut down and cool itself without any human intervention or external power or control rods inserting, to India's PFBR fast reactor that's been under construction for the past 18 years.

Only the Russians have successfully operated fast reactors commercially, via the BN-350 and BN-600. In general, fast neutron reactors are considered more complex and expensive to build and operate than traditional water-cooled thermal neutron reactors.

We once thought uranium was very scarce, and so we put a lot of money into fast breeder reactors. Then it turned out that there's lots of uranium and fast reactors are largely on hold internationally. France just announced the cancellation of its national fast reactor program (ASTRID). Russia delayed their next BN fast reactor, saying their VVER (slow neutron) reactors are bout 4x cheaper. China and India are pushing forward. The US shut down its last fast reactors (EBR-II and FFTF) in the early 1990s. Many nuclear startup companies are now exploring options to get back into fast reactors.

The US DOE is putting forth a major project to build a new fast reactor very similar to the FFTF, but in Idaho instead of Washington and with metal fuel rods instead of oxide ceramic ones. The project is called the Versatile Test Reactor (VTR) and is actively discussed in current nuclear news.