Here is a mathematician's domain. Most of filters are using convolution
matrix. With the Convolution Matrix filter, if the fancy takes you, you
can build a custom filter.

What is a convolution matrix? It's possible to get a rough idea of it
without using mathematical tools that only a few ones know. Convolution
is the treatment of a matrix by another one which is called
“kernel”.

The Convolution Matrix filter uses a first matrix which is the Image to
be treated. The image is a bi-dimensional collection of pixels in
rectangular coordinates. The used kernel depends on the effect you want.

GIMP uses 5x5 or 3x3 matrices. We will consider only 3x3 matrices, they
are the most used and they are enough for all effects you want. If all
border values of a kernel are set to zero, then system will consider it
as a 3x3 matrix.

The filter studies successively every pixel of the image. For each of
them, which we will call the “initial pixel”, it
multiplies the value of this pixel and values of the 8 surrounding
pixels by the kernel corresponding value. Then it adds the results,
and the initial pixel is set to this final result value.

A simple example:

On the left is the image matrix: each pixel is marked with its value.
The initial pixel has a red border. The kernel action area has a green
border. In the middle is the kernel and, on the right is the convolution
result.

Here is what happened: the filter read successively, from left to right
and from top to bottom, all the pixels of the kernel action area. It
multiplied the value of each of them by the kernel corresponding value
and added results. The initial pixel has become 42:
(40*0)+(42*1)+(46*0) + (46*0)+(50*0)+(55*0) + (52*0)+(56*0)+(58*0) = 42.
(the filter doesn't work on the image but on a copy). As a graphical result,
the initial pixel moved a pixel downwards.

Design of kernels is based on high levels mathematics. You can find
ready-made kernels on the Web. Here are a few examples: