Being an engineer, I have sometimes heard of
Topological filters and in particular ultrafilters. They seem to have some meaning regarding partial ordering of sets but be of a more pure math origin than I am used to.
Then there is the signal processing meaning of filter - a function which is applied when it is convolved with a "signal" (another function):
$$(f*g)(t) = \int_{-\infty}^{\infty}f(\tau)g(\tau-t)dt$$
I suspect these two usages for the word "filter" may be distinctly different with no intuitive connection. But maybe there is some connection that my lack of studies in pure mathematics does not allow me to see?