For purposes of intuition, I've found the following reformulation of the definition useful. A subset $S$ of $\kappa$ is stationary if and only if, for any countably many finitary operations on $\kappa$, say $f_n:\kappa^{r_n}\to\kappa$, there is an $\alpha\in S$ closed under all the $f_n$'s. In the terminology of universal algebra, this says that every algebra with underlying set $\kappa$ and with countable type has a subalgebra that is a member of $S$. By using Skolem functions, one can also rephrase the definition in model-theoretic terms: Any structure with universe $\kappa$ for a countable language has an elementary submodel whose universe is a member of $S$. (That bring us into the general neighborhood of the somewhat deeper comments in Andres Caicaedo's answer.)

For purposes of intuition, I've found the following reformulation of the definition useful. A subset $S$ of $\kappa$ is stationary if and only if, for any countably many finitary operations on $\kappa$, say $f_n:\kappa^{r_n}\to\kappa$, there is an $\alpha\in S$ closed under all the $f_n$'s. In the terminology of universal algebra, this says that every algebra with underlying set $\kappa$ and with countable type has a subalgebra that is a member of $S$. By using Skolem functions, one can also rephrase the definition in model-theoretic terms: Any structure with universe $\kappa$ for a countable language has an elementary submodel whose universe is a member of $S$. (That bring us into the general neighborhood of the somewhat deeper comments in Andres Caicedo's answer.)