What does a stay in a civil matter mean?

It is essentially no different than in a criminal matter/proceeding: when a stay issues, some prior legal writ or mandate ceases in operation until the or matter or challenge which gave rise to the stay is resolved.

Often a stay might be issued, for example, to allow for a contingency of some kind. For example, a judgment for damages could be stayed for a time during which the judgment debtor would be allowed to fulfill some alternative relief in favor of plaintiff . . . such as, for example, the return of misappropriated property--in the absence of occurrence of which, or upon the fulfillment of which, the stay would be lifted or would lift of its own accord and no longer have effect.

Added: Short answer: In a legal context, a stay is a court order preventing further action until a future event occurs, or the order is lifted.