Ask how thick water is and the question has one answer: its viscosity is a fixed number at a given temperature, no matter how gently or hard you stir it. Water is Newtonian. Ask how thick paint is, and the honest answer is a question back: how fast are you moving it? Paint is thin under a brush and thick on the wall. That is non-Newtonian behaviour, and for a whole class of everyday fluids, a single viscosity number is incomplete, and so misleading.
Viscosity, defined by shear
Newton pictured a fluid as stacked layers and defined viscosity as the ratio of the shear stress you apply to the shear rate it produces.1 For a Newtonian fluid that ratio is constant: push twice as hard, it flows twice as fast, and the viscosity is the same number either way. Plot shear stress against shear rate and you get a straight line through the origin whose slope is the viscosity. One line, one number.
When the line bends
For a non-Newtonian fluid the line is not straight, and the "viscosity" you read depends on where on the curve you measured. The common kinds are worth naming:
Shear-thinning (pseudoplastic) fluids thin as you push harder — paint, polymer solutions, ketchup, blood. The most common non-Newtonian behaviour, and the reason a VI-improved oil can be thinner in a bearing than on the bench. Shear-thickening (dilatant) fluids do the opposite and stiffen under stress — a cornstarch-and-water suspension is the kitchen example. Thixotropic fluids thin over time under shear and rebuild at rest, so their history matters. And a Bingham plastic will not flow at all until a yield stress is exceeded — toothpaste holds its shape until you squeeze.
Report "the viscosity of the grease is X" and you have said almost nothing, because X is only true at the one shear rate you happened to measure. A pump, a bearing, and a settling tank each impose wildly different shear rates, and the grease has a different effective viscosity in each. For a non-Newtonian fluid, a viscosity value with no shear rate attached — and for thixotropic fluids, no time or history — is not a measurement anyone can use.
Measuring a fluid that moves
This is why a capillary viscometer, which measures at essentially one shear rate, is fine for a Newtonian oil but not for a paint. A rotational viscometer or rheometer shears the fluid at a range of rates and traces the whole flow curve, which is the description a non-Newtonian fluid needs.2 Density and its pairings still tell you what a fluid is; for how it flows, the shear rate is part of the answer.
Before you trust a viscosity number, ask whether the fluid is Newtonian. If it is — most light oils, solvents, fuels — one number is enough. If it is not, one number is a headline with the story missing, and the shear rate is the story.