A stay of execution is a time period in which a judgment of execution is prohibited from being carried out. It may arise through an agreement between parties or by law, usually on condition of entering bail or security for the money. In landlord-tenant law, a stay of execution may be given by a judge to a tenant ordered to vacate the premises. Such a stay is usually based upon proof that the tenant will suffer hardship without an extension and has failed to obtain other housing after diligent efforts. The tenant will usually be required to pay rent (or what the judge thinks is a fair value of the apartment) until they move. Laws vary by locality, so local law should be consulted.
For example, a couple of the Oklahoma bombing victims filed a motion asking the federal court governing Terre Haute prison to stay McVeigh's execution pending a civil suit the victims filed - claiming that McVeigh was an indispensable witness in that lawsuit, which was in the discovery phase.