Mix two parts of corn starch with one part of water: rejoice! you now have a non-Newtonian fluid…
A normal fluid is usually described by a handful of properties: density, velocity, pressure, viscosity. That's five dynamical quantities: one density, three velocities (in 3D), and one pressure - plus viscosity which is taken non-dynamical, a constant property of the fluid. Viscosity is a measure of the resistance of a fluid to flow: honey is more viscous than water since it pours more slowly; it's "stiffer" and hence has a higher constant value for its viscosity parameter.
To predict the evolution of a flow, you need five equations from physics - for your original five properties of the fluid. Three come from Newton's second law applied to a fluid (you know, the good old force equal mass times acceleration in 3D); another comes of the fact that you are hopefully not allowed to create fluid out of nothing (mass conservation); and the last is known as an Equation of State: it tells you how the pressure and density in the fluid are related. This last equation depends on the type of fluid. It is for example different for water from that for Hydrogen gas (yes, a gas is very much a fluid: anything that flows is called a fluid). At the end of the day, you end up with a set of equations that you can try to solve: that is if you're really really (and I mean really) lucky. Most of the time, you just stare at them in despair and ponder alternate less glamorous fallback careers: the equations are just too nasty to solve by hand! So, you use a computer to determine numerical predictions… and even those blow up in your face very quickly because of a fun thing known as turbulence - the same thing that you experience in an airplane while trying to drink a cup of coffee. That's Newtonian fluid for you. You now know why I avoid fluid dynamics in my research whenever possible.
Now that you've suffered enough reading through equation counting, time to come back to non-Newtonian fluids and have some actual fun. In a Non-Newtonian fluid, the viscosity is NOT a constant… It depends on the pressure! The fluid has an additional equation relating the changing pressure to its changing viscosity - which is now dynamical… A non-Newtonian fluid becomes stiffer when you apply strong pressure: it behaves like a solid! this same stuff flows smoothly like syrup when under low pressure… As if a chameleon, the Non-Newtonian fluid changes its viscosity according to its environment! Now, I already told you that you can't solve most Newtonian fluid problems without a computer. So, forget about tackling non-Newtonian stuff even with loads of charisma and skills. This is an area of physics that is still currently under active research and investigation. In the meantime, check out the two videos attached to this post to see what we know so far about these very very interesting substances. The first video shows what happens when you bring together a group of intoxicated people and a pool of corn starch. The second video will bring out in you new emotions: anything from disgust to terror.