Any polyatomic gas is a greenhouse gas. If it's airborne and has three or more atoms, it qualifies.
Low energy infrared photons (like those emitted by a body at 300K) can cause bonds to bend side to side in a flapping motion.
Oxygen and nitrogen are diatomic molecules. They can stretch, but there's no way they can bend because there are only two atoms. So they're transparent to IR emitted from the ground and are not greenhouse gases. Molecules that can bend need three atoms or more, like carbon dioxide, which gets hit by an infrared photon and moves like a bird flapping its wings before reemitting it. H2O is also a greenhouse gas but its long term atmospheric concentration is stable over the long term and doesn't rise year over year. Methane is a potent gas because it's tetrahedral and its single bonds are easier to flex than e.g. the double bonds in CO2.
HFCs and CFCs also have tetrahedral shapes with single bonds, but they're more potent greenhouse gases than methane, because the fluorine and chlorine atoms distort the charge concentration and give the molecule a dipole moment that makes it better at scattering photons. They also provide it with more possible bending motions.
I'm not sure why this article is talking about illegal fluorotrichloromethane being a greenhouse gas. It's illegal because it destroys stratospheric ozone. Gram for gram, sarin is an extremely potent greenhouse gas, but that's not why it's illegal.