What is the difference between convolution filter, low-pass filter, and median filter?
A median filter is most certainly not a "blur" filter, purely on the basis that it tends to preserve edges. Edges are abrupt transitions of brightness and therefore that information is encoded in the high frequencies of the spectrum. Incidentally those high frequencies are the ones that low-pass filters suppress the most, leading to that "blurry" appearance because only the form of the depicted objects is retained rather than their details.

Convolution can only be used to represent linear time invariant systems. That is, systems whose relationship between the input and the output is a linear combination and therefore proportional inputs produce proportional outputs. Furthermore, this relationship does not change with respect to time.

A median operator works by sorting the values of the $M \times N$ mask of pixels surrounding some $i,j$ pixel and then assigning their median (the pixel value that happens to lie at the midpoint of the range of values) to the $i,j$ pixel.

This operation, of sorting and then selecting the median, cannot be expressed as a linear combination of the pixel values within the $M \times N$ mask, hence the name "non-linear filter".

Hope this helps.