This answer is in complement of HBR's answer, following the discussion we had in the comments.
I took from him the structure of it and colours so to keep consistency and make it easier for those who read both. Any mistakes are mine.
We have the following problem:
$$\frac{\partial u}{\partial t}+v\color{red}{\frac{\partial u}{\partial x}}-\nu\color{blue}{\frac{\partial^2u}{\partial x^2}}=0 \tag{*}$$
were $u$ is a quantity that diffuses with diffusion coefficient $\nu>0$ and is advected by velocity $v>0$.
Imagine we discretise the equation $(*)$ according to the following scheme (upwind for the advection term, central difference for the diffusion term) on a 1D equispaced mesh:
$$\color{red}{\frac{\partial u}{\partial x} = \frac{u_i-u_{i-1}}{\Delta x}}$$
$$\color{blue}{\frac{\partial^2 u}{\partial x^2}\approx \frac{u_{i-1}-2u_i+u_{i+1}}{\Delta x^2}}$$
Giving the following numerical approximation:
$$\frac{\partial u}{\partial t}+v\color{red}{\frac{u_i-u_{i-1}}{\Delta x}}-\nu\color{blue}{\frac{u_{i-1}-2u_i+u_{i+1}}{\Delta x^2}}=0 \tag{**}$$
Taylor expansion for $u_{i\pm 1}$ around $u_i$ is
$$u_{i\pm 1} =
u_i
\pm \frac{\partial u}{\partial x}\Delta x
+\frac{1}{2}\frac{\partial^2u}{\partial x^2}\Delta x^2
\pm\frac{1}{6}\frac{\partial ^3 u}{\partial x^3}\Delta x^3
+ \frac{1}{24}\frac{\partial^4u}{\partial x^4}\Delta x^4
\pm\frac{1}{120}\frac{\partial^5 u}{\partial x^5}\Delta x^5
+\mathcal{O}(\Delta x^6)$$
If we use this Taylor expansion in $(**)$ find out what equation $(**)$ solves, we find:
$$\frac{\partial u}{\partial t}+v
\left[\color{red}{
\frac{\partial u}{\partial x}
-\frac{1}{2}\frac{\partial^2 u}{\partial x^2}\Delta x
+\mathcal{O}(\Delta x^2)
}\right]
-\nu\left[\color{blue}{
\frac{\partial^2u}{\partial x^2}
+\frac{1}{12}\frac{\partial^4u}{\partial x^4}\Delta x^2
+\mathcal{O}(\Delta x^4)
}\right]=0 $$
We can say that when using the scheme $(**)$ we approximate the following equation exactly (up to an error $\mathcal{O}(\Delta x^2)$):
$$\frac{\partial u}{\partial t}+v\color{red}{\frac{\partial u}{\partial x}}-\left(\nu+\color{red}{\frac{v\Delta x}{2}} -
\color{blue}
{\frac{\nu\Delta x^2}{12}\frac{\partial^2}{\partial x^2}}
\right)\color{blue}{\frac{\partial^2u}{\partial x^2}}=\color{red}{\mathcal{O}(\Delta x^2)}$$
where $\frac{\partial^2}{\partial x^2}$ is not a quantity, but an operator.
The lowest order term introduced by the discretization of the advection term is the numerical diffusion $\frac{v\Delta x}{2}\frac{\partial^2u}{\partial x^2}$, which would be unimportant just if
$$
\color{red}
{\frac{v \Delta x}{2 \nu} \ll 1} \tag{***}
$$
The quantity $\frac{v \Delta x}{\nu}$ is known as the grid Péclet number. $(** $$*)$ is rarely true, so the lowest order term introduced by the discretization of the advection term is rarely unimportant.
The lowest order term introduced by the discretization of the diffusion term is $\frac{\nu\Delta x^2}{12}\frac{\partial^4u}{\partial x^4}$, which is unimportant if $\frac{v\Delta x}{2}\frac{\partial^2u}{\partial x^2}$
$$\color{blue}{\frac{\nu\Delta x^2}{12}\frac{\partial^4u}{\partial x^4}} \ll \color{red}{\frac{v\Delta x}{2}\frac{\partial^2u}{\partial x^2}}$$
$$\frac{\nu\Delta x}{6x_0^2} \ll v \tag{****}$$
where $x_0$ is some characteristic length. $(**$$**)$ is usually true, so the lowest order term introduced by the discretization of the diffusion term is usually unimportant.