Quantum groups are not really groups, and are only a little bit quantum. What they in fact provide for us is a beautiful and compact framework for tweaking the representation theory of classical groups like GL_2 to yield payouts in low dimensional topology.
In this colloquium, I'll explain one of Drinfeld's early constructions of quantum groups via monodromies -- i.e. ordinary differential equations -- on CP^1, and then a more algebraic take on them due to Drinfeld, Jimbo and Kohno. I'll then explain a construction of Reshetikhin and Turaev that recovers essentially all known polynomial invariants of knots from this framework.
Finally, I'll highlight a remarkable property of quantum groups: they admit a kind of Fourier transform familiar to abelian groups, but lacking for non-abelian groups like GL_2. It turns out this is also a reflection of their role in low dimensional topology.