The difference between these classifications of fluids is in how they resist the flow versus a Newtonian fluids.
For fluids like water an increase in shear force in linearly proportional to the rate of shear deformation. To put it in the context of a pipe system, the harder a pump pushes (or pulls) a Newtonian fluid, the faster it flows. This is ignoring turbulent friction and other things, but in an idealized situation we can say that doubling the power of the pump would double the flow rate in the pipe.
To give two examples of what a non-newtonian fluid would do there: ketchup is a Bingham plastic, which acts solid under low stresses and then flows if that stress passes a threshold. Imagining the pipe and pump are filled with stationary ketchup and the pump is turned off, we can think about what would happen if we turned a knob that slowly ramped up power to the pump. At first the ketchup wouldn’t move, and the pump would just heat up the ketchup from wasting the work done. Then, turning the know further, it will eventually reach the break point and the fluid will begin to flow like a Newtonian fluid, and linearly proportional too. But if the power is decreased again such that the stress is below that yield stress the fluid will become stationary again.
Next example would be Oobleck, or a shear thickening fluid, which when we construct the same scenario, will flow “easily” at low power through the pump, and a low flow rate. As the power/flow rate/shear stress is increased then the resistance to the flow will increase dramatically and there will be incredible diminishing returns to the increase in flow rate for the increase in pumping power.
In reality there are a lot more asterisks and corollaries, like the fact that I think that oobleck would seize up at an flow rate that would be too slow to be a worthwhile demonstration.
