Intuitive definition for Newtonian fluid 
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*What is an intuitive definition of a Newtonian fluid?

I know that it is a type of fluid in which shear stress is linearly proportional to its slop of v and y. I want a definition without any mathematical proof.
 A: Since you're looking for an intuitive picture, it may help to draw analogy between Hookean solids (solids whose elastic behaviour is well captured by the Hooke's law) and Newtonian fluids. For example both fall into a first-order linear approximation response theory and therein lies the intuition: The response of the solid (deformation-strain) or the fluid (rate of change of deformation over time-strain rate) to the applied stress differs only by a proportionality constant, meaning the system (and more precisely its behaviour to external influences) will not undergo permanent changes (no non-linear effects), no matter how quickly, weakly or strongly it is being sheared. This in turn means that we can define a characteristic constant of stiffness for the elastic behaviour of the solid and similarly a characteristic dynamic viscosity for the fluid. 
Now in the case of solids and their elastic behaviour, we apply a stress to them and study their mechanical response, e.g. how stiff they are, how much do they elongate, whereas for fluids, when one applies a shearing stress, their response is to move (as they have no stiffness) which is determined by their viscosity.
Now you may have also heard of viscoelastic materials, which as the name suggests, exhibit both viscous (they can flow) and elastic (they can also act stiff) characteristics when subject to a shear stress. Depending on the rate with which we are shearing the material its behaviour varies between that of the elastic solids and viscous fluids. For example a hysteresis stress-strain curve can often be observed, hinting at the fact that unlike Hookean solids, viscoelastic materials do dissipate energy when under load. Good example of such materials are honey and silly putty.  
Let's wrap up by reminding the definitions:


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*Hookean solid: $\sigma = \gamma \epsilon$ ($\sigma$ denotes the applied stress, $\gamma$ the elastic modulus and $\epsilon$ the deformation). These are materials that exhibit linear elasticity, and deform proportionally to the applied load. Note: Only an idealisation, as in reality no material can be indefinitely compressed or stretched, instead there are thresholds of deformation beyond which the material will undergo permanent change (e.g. broken covalent bonds in a polymer chain) and will be unable to return to its undeformed state when the load is removed.

*Newtonian fluid: $\sigma = \eta \frac{d\epsilon}{dt}$ ($\eta$ denotes the viscosity of the material and $\frac{d\epsilon}{dt}$ the strain rate). In shear experiements, all such fluids under constant pressure and temperature conditions show a constant resistance to flow, i.e., there is a linear relationship between the viscous stress and the strain rate. Examples: air, water, milk... Note: Again only an idealisation as e.g. the viscosity of a real fluid does depend on the velocity field of its flow. Moreover a simple characteristic viscosity constant cannot be defined, as in most real cases the viscosity is a function (often non-linearly so) of the stress or strain rate. For example in rheology, we speak of shear-thinning or thickening, referring to cases where as a function of shear rate the viscosity of the material decreases and increases respectively (e.g. for paint). 

Encouraged further reading:
Newtonian Fluid, viscous stress tensor, viscoelasticity, linear response theory.
A: First, some equations!
A Newtonian fluid is a fluid for which 
$$\tau = \eta(T) \cdot \dot \gamma$$
where $\tau$ is the shear stress, $\eta$ is the viscosity, $T$ is the temperature and $\dot \gamma$ is the shear rate. 
$\tau$ represents the force per unit area that the flowing liquid exerts on a surface, in the direction parallel to the flow.
$\dot \gamma$ is in general a tensor $\dot \gamma_{ij}$ and it is defined as
$$\dot \gamma_{ij} = \frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}$$
where $v_i$ are the components of the velocity field of the fluid.
If we a "sandwich" of two plates containing a fluid layer of small thickness $h$, with one plate fixed an another moving with velocity $v$, we will simply have 
$$\dot \gamma = \frac v h$$
(because since $h$ is small we can make the approximation that the velocity of the fluid is approximatively constant and equal to the velocity of the plate).
The shear stress would then be
$$\tau = \eta(T) \frac v h$$
For a non-newtonian fluid, the viscosity $\eta$ also depends on the shear rate:
$$\tau = \eta(T, \dot \gamma) \cdot \dot \gamma$$
so that for the sliding plates case we have
$$\tau = \eta \left(T, \frac v h \right) \cdot \left( \frac v h \right)$$
Ok, nice equations: but what does this all mean?
The viscosity $\eta$ is a measure of the resistence of the fluid to an applied shear stress: if a fluid has a large viscosity, even a small shear rate will generate a large shear stress, i.e. the force exerted on the surface will be large and it will be difficult to make the liquid flow (for example inside a pipe). 
For Newtonian fluids, it doesn't matter how fast you apply the stress: the viscosity, i.e. the response to the stress, will be constant. Examples of Newtonian fluids are water, gasoline and alcohol.
For non-Newtonian fluids, it is important to consider how fast we apply the stress: there are some fluids, called shear-thickening, for which viscosity increases with stres. For others, called shear-thinning, viscosity decreses with stress.

For example, ketchup is a shear-thinning liquid: you may have noticed if you've ever had difficulties in making it come out of the bottle. What do you do to succeed in the task? You smack the bottle, and hard: this way, the shear rate increases (think about the relative velocity between the ketchup and the internal walls of the bottle) and viscosity decreases.
Examples of shear-thickening fluids are quicksand and the famous "oobleck" (starch+water): the faster you apply the stress, the higher the viscosity. This is why getting agitated and struggling into quicksand will only make it worse, and this is also why oobleck looks almost solid when put on a fast-vibrating speaker cone.

[Source of the figure above].
