Stationary sets are exactly the positive-measure sets with respect to the club filter, which is a very natural measure on the subsets of $\omega_1$ or on higher cardinals with uncountable cofinality. Thus, they provide a notion of largeness, which could be used to provide a notion of large subalgebras.

For example, the non-stationary sets in $\omega_1$ form an ideal NS, whose quotient $P(\omega_1)/NS$ is a highly studied Boolean algebra in set theory. The properties of this algebra, such as whether it is precipitous or saturated, are connected with deep concepts in set theory, including large cardinals.

But you emphasized basic applications, so let me tell you one way of thinking about stationarity.

Theorem. A set $S\subseteq\omega_1$ is stationary if and only if for every algebra $\langle\omega_1,f_1,\ldots,f_n,\ldots\rangle$ on $\omega_1$, allowing countably many functions of finite arity, there is an element $\gamma\in S$ such that $\gamma$ is a subalgebra.

Proof. The point is that every algebra has a closed unbounded collection of $\gamma$ that are closed under the functions of the algebra. And conversely, for every closed unbounded set, there is a function whose closure points are in the club: the function mapping every element to the next element of the club. Since a set is stationary just in case it has elements of every club set, one can also say that $S$ is stationary just in case it provides subalgebras for every algebra. $\Box$

This theorem is often used in set theory not in the context of rings, but rather by expanding a given structure by Skolem functions, so that the subalgebras of the expansion correspond to elementary substructures of the original structure. In this way, the stationary sets are those that provide elementary substructures of any given algebra.

The theorem generalizes in a very attractive and natural way to the concept of generalized stationary, in the structure $P_{\omega_1}(X)$, consisting of the countable subsets of $X$. Here, a set $S\subseteq P_{\omega_1}(X)$ is defined to be stationary just in case for every algebra on $X$, there is a subalgeba in $S$.

Stationary sets are exactly the positive-measure sets with respect to the club filter, which is a very natural measure on the subsets of $\omega_1$ or on higher cardinals with uncountable cofinality. Thus, they provide a notion of largeness, which could be used to provide a notion of large subalgebras.

For example, the non-stationary sets in $\omega_1$ form an ideal NS, whose quotient $P(\omega_1)/NS$ is a highly studied Boolean algebra in set theory. The properties of this algebra, such as whether it is precipitous or saturated, are connected with deep concepts in set theory, including large cardinals.

But you emphasized basic applications, so let me tell you one way of thinking about stationarity. This amounts to a fundamentally algebraic characterization of stationarity.

Theorem. A set $S\subseteq\omega_1$ is stationary if and only if for every algebra $\langle\omega_1,f_1,f_2,\ldots\rangle$ on $\omega_1$, allowing countably many functions of finite arity, there is an element $\gamma\in S$ such that $\gamma$ is a subalgebra.

Proof. The point is that every algebra has a closed unbounded collection of $\gamma$ that are closed under the functions of the algebra. And conversely, for every closed unbounded set, there is a function whose closure points are in the club: the function mapping every element to the next element of the club. Since a set is stationary just in case it has elements of every club set, one can also say that $S$ is stationary just in case it provides subalgebras for every algebra. $\Box$

The theorem generalizes to higher cardinals. This theorem is often used in set theory not in the context of rings, but rather by expanding a given structure by Skolem functions, so that the subalgebras of the expansion correspond to elementary substructures of the original structure. In this way, the stationary sets are those that provide elementary substructures of any given algebra.

The theorem also generalizes in a very attractive and natural way to the concept of generalized stationary, in the structure $P_{\omega_1}(X)$, consisting of the countable subsets of $X$. Here, a set $S\subseteq P_{\omega_1}(X)$ is defined to be stationary just in case for every algebra on $X$, there is a subalgeba in $S$.