There is a definition of a compact set, which involves subsequences. Namely:
Set $A \subset R^d$ is called compact, if for every sequence ${x_n}$ with $x_n \in A$ there exists a subsequence with a limit belonging to A.
This definition is a bit counter-intuitive to me. I get for example, what compactness means in terms of "closed and bounded set". It's pretty clear that if a set is not unbounded (not an infinitely stretching ball) and every sequence of points is fully included inside that ball (including the limit of this sequence) then we call it compact. This way it becomes clear why an 'open' ball can't be compact, namely, since its limit might lie on a surface of the ball, which does not belong to the set itself.
But I can't get what the definition through subsequences is supposed to mean.
We can directly relate the subsequential notion of compactness to the standard open cover definition of compactness. It might be useful to note first that compactness is, at root, a generalisation of finiteness. Finite sets tend to be easy to prove things about because we can work by exhaustion: each element can be individually inspected, if necessary, to prove or disprove a hypothesis. Compactness generalises that to infinite sets by giving us a way to reduce things to the finite case: if we can say something about an open set then compactness says we only need to inspect finitely many such open sets to obtain a result about whatever the set covers.
So: compactness is, by definition, the notion that every open cover of a set has a finite open subcover. In some[1] topologies we can directly connect this to sequences: around each element of a sequence we can find/construct an open set. If we take those sets large enough then we have an open cover of the space. If there is a finite subcover, then the space is compact. But a finite subcover is, by our construction, a subsequence of our original sequence. This has to be true for every sequence in the space though: if we can find a sequence where we can't construct an open cover of the space using it, then the space cannot have an open cover. Hence sequential compactness.
[1] Not all; hence why subsequential compactness is less useful as a definition.