I propose the following way to introduce filters on sets to beginning students. (I am writing a book which contains this intro now.) You are welcomed to comment whether this is a good exposition and how to make it even better.

We sometimes want to define something resembling an infinitely small (or infinitely big) set, for example the infinitely small interval near $latex 0$ on the real line. Of course there are no such set, just like as there are no natural number which is the difference $latex 2 – 3$. To overcome this shortcoming we introduce whole numbers, and $latex 2 – 3$ becomes well defined. In the same way to consider things which are like infinitely small (or infinitely big) sets we introduce

filters.An example of a filter is the infinitely small interval near $latex 0$ on the real line. To come to infinitely small, we consider all intervals $latex \left( – \varepsilon ; \varepsilon \right)$ for all $latex \varepsilon > 0$. This filter consists of all intervals $latex \left( – \varepsilon ; \varepsilon \right)$ for all $latex \varepsilon > 0$ and also all subsets of $latex \mathbb{R}$ containing such intervals as subsets. Informally speaking, this is the greatest filter contained in every interval $latex \left( – \varepsilon ; \varepsilon \right)$ for all $latex \varepsilon > 0$.

[A formal definition of a filter on a set goes here.]