Ulam matrices appeared in Ulam's doctoral thesis, written in Polish and presented at the Technical University of Lwów in 1933. The results of the dissertation were published in two articles [Fund. Math. 14 (1929), 231–233, and Fund. Math. 16 (1930), 140–150]. The first one is written in English. The second one is written in German, and it is the one in which Ulam matrices appear; a free copy of it can be found here.
Ulam matrices are used to prove the theorem below, which as explained by the remark following it, implies that every stationary subset of a successor cardinal $\kappa$ can be partitioned into $\kappa$ disjoint stationary sets.
The theorem and the remark following it are taken from Kunen's textbook "Set Theory" (1980 edition, p.79).
Theorem (Ulam). If $\kappa$ is a successor infinite cardinal, and $\mathcal{I}$ is a $\kappa$-complete ideal on $\kappa$ such that each singleton is in $\mathcal{I}$, then there are disjoint $X_{\alpha}$ for $\alpha<\kappa$ such that $X_\alpha\notin \mathcal{I}$ for each $\alpha$.
Remark. If $\kappa$ is as above, then whenever $S$ is any stationary subset of $\kappa$, $S$ maybe partitioned into $\kappa$ disjoint stationary subsets since we can apply Ulam's above theorem to the ideal $\mathcal{I}$, where $\mathcal{I} =\{X \subseteq \kappa$ : $X \cap S$ is nonstationary}. $\mathcal{I}$ is $\kappa$-complete since the intersection of less than $\kappa$-many closed unbounded sets is closed unbounded.
History of the concept of stationarity. According to Thomas Jech's textbook "Set Theory" (Millenium edition, p. 105) and this article by Jean Larson (footnote 213 on page 82) the notion of stationarity was explicitly introduced by Gérard Bloch in 1955, but the concept is implicit already in the work of Paul Mahlo (1911).
Two relevant MO answers are this MO-answer by Péter Komjáth and This MO answer by Joel David Hamkins.
