If are an non-citizen of the United States and are facing the deportation, otherwise defined as the process of expelling a foreigner from the country, you will likely seek the the assistance of a lawyer to get permission to remain in the country legally. This postponement is called a "stay of deportation" or "stay of removal".
There are various government agencies involved in the immigration process. Each agency handles requests for a stay of deportation and appeals differently.
Typically, a stay of deportation would be requested while you are appealing your case.