Nets are an important generalization of sequences. Equivalently, convergence of sequences and the like can be formulated using filters instead of nets.
As for the notion of sub-sequence, it can be expressed in terms of nets by taking a cofinal subset of the directed indexing set. But how do you express sub-sequences in terms of filters?
For every net $\langle x_\alpha\rangle_{\alpha\in A}$, you call a set of the form $\{x_\beta : \beta\in A, \;\beta\succeq\alpha\}$ a tail-set. The tail-sets form a filter base, and the filter generated by the tail-sets is the eventuality filter of the net. The eventuality filter of a subnet will be larger under set inclusion then. The characterization of compactness that every net has a convergent subnet then turns into the corresponding argument that for every filter there exists a larger filter that converges. Equivalently, maximal filters,always converge and these are just the ultrafilters.
To get an actual correspondence, we have to show that every filter is the eventuality filter of some net. Take a filter $\mathcal{F}$. Ordering $X\times\mathcal{F}$ by $(x,B)\succeq(y,B')$ iff $B\subseteq B'$ we get a directed set and with this directed set, $(x,B)\mapsto x$ becomes a net with eventuality filter $\mathcal{F}$.
A great source for material on the relationship between nets and filters is the Handbook of Analysis and its Foundations by Eric Schechter.