Put very simply, a Newtonian fluid is something (like water) that remains perfectly fluid despite stirring the hell out of it. The thicker a fluid gets, it gets harder to stir it which is kind of what viscosity is. But you can still stir it.
A non-Newtonian fluid, however, is a bit weird. When you stir it, you create a hole. This whole will fill itself up over time. This viscosity increases like crazy when you stir it. Good examples are Silly Putty and (quick) sand. (It’s worth pointing out that Silly Putty is far safer than quick sand when it comes to demonstrating non-Newtonian fluid dynamics at home.)
Also worth noting is that there are some non-Newtonian fluids that get smoother when you stir them. You may have seen it with certain kinds of paint which start out extremely thick but ‘thin’ themselves when stirred.