What does it mean in image processing when a filter is called non-linear?
A filter F is called "linear", iff for any scalars $c_1$, $c_2$ and any images $I_1$ and $I_2$: 

$F\left(c_1\cdot I_1+c_2\cdot I_2\right)=c_1\cdot F\left(I_1\right)+c_2\cdot F\left(I_2\right)$

This includes:


Derivatives
Integrals
Fourier transform
Z-Transform
Geometric transformations (rotate, translate, scale, warp)
Convolution and Correlation
the composition of any tuple of linear filters (i.e. applying some linear filter to the output of another linear filter $F(G(I))$)
the sum of the result of any two linear filters (i.e. the output of one filter, added pixel by pixel to the output of another filter $F(I) + G(I)$)


and many others.

Examples of non-linear filters are:


the square, absolute, square root, exp or logarithm of the result of any linear filter
the product of the result of any two linear filters (i.e. the output of one filter, multiplied pixel by pixel with the output of another filter $F(I)\cdot G(I)$)
morphological filters
median filter