The notion of an unbounded family can be made very general. Suppose that $R$ is any relation on a set $X$. We could define a subset $A \subseteq X$ to be unbounded if for each $x \in X$ there is an $a \in A$ such that $a \not\mathrel{R} x$ -- that is, no single $x \in X$ serves as an upper bound for $A$. Of course, there are situations where this notion may not be too meaningful.
Example. Consider any partially ordered set $( P , \leq )$ containing two elements $a, b$ which have no common upper bound. Then $\{ a , b \}$ is an unbounded subset of the partial order, irrespective of the lengths of chains that might exist in $P$. In particular, if $\kappa$ is any cardinal, then the partial order consisting of two disjoint copies of $\kappa$ under the usual order will be of this type.
To make this notion somewhat more meaningful we may assume that given any two $x , y \in X$ there is a $z \in X$ such that $x \mathrel{R} z$ and $y \mathrel{R} z$ both hold.
An important set-theoretic/combinatorial notion related to this is the bounding number. Consider the family $\omega^\omega$ of all functions from $\omega$ to itself. We define a relation $\leq^*$ on $\omega^\omega$ by declaring $f \leq^* g \quad \Leftrightarrow \quad ( \exists N )( \forall n \geq N ) ( f(n) \leq g(n) ),$ (i.e., the relation $f(n) > g(n)$ holds for at most finitely many $n$). It is not too difficult to show that $( \omega^\omega , \leq^* )$ has the property described above: given $f,g \in \omega^\omega$ define $h(n) = \max \{ f(n) , g(n) \}$.
According to the above, an unbounded family is then a subset $B \subseteq \omega^\omega$ such that for each $g \in \omega^\omega$ there is a $f \in B$ such that $f \nleq^* g$, i.e., $f(n) > g(n)$ holds for infinitely many $n$. Clearly $\omega^\omega$ is itself an unbounded family, and so we may ask about the minimum cardinality of such a set. This is called the bounding number, and is usually denoted by $\mathfrak{b}$. It is easy to show that $\aleph_1 \leq \mathfrak{b} \leq \mathfrak{c} = 2^{\aleph_0}$, but the exact value of this cardinal, as well as how it sits between $\aleph_1$ and $\mathfrak{c}$, is independent from the ZFC. Much has been written about this and other so-called cardinal characteristics of the continuum, many of which can be given a similar definition. (Perhaps surprisingly the value $\mathfrak{b}$ puts some bounds on some notions related to Lebesgue measure are Baire category.)