Let $\kappa$ be a cardinal (of uncountable cofinality). A subset $S \subseteq \kappa$ is called stationary if it intersects every club, i.e. closed unbounded subset of $\kappa$. Now my question is basically just: Why do we care about stationary sets? I know some statements, which are independent from ZFC, for example the diamond principle, which involve them, but what was or is the motivation for studying them? Please don't just give an overview of the results, because they can be found in every textbook on cardinals.
Also, what is your intuition for stationary sets? I think clubs are very easy to grasp; they just contain big enough ordinals. Perhaps stationary sets follow the same idea, but in a "second order"; they contain enough big enough ordinals?