This ex parte application arises when you are facing lockout because your default has been taken, default judgment has been entered against you, a writ of possession has been issued against you, and/or the Sheriff has posted a Notice to Vacate on your door, giving you only a few days to be moved out. "Staying execution" in this case means stopping the Sheriff from completing the lockout, to give the court time to hear your motion for relief from default or to vacate the erroneous clerical or judicial error. It's a hold-everything order, suspending activity, while shortening the time for your motion to vacate or relief from default so that it is quickly heard and not a month from now.

If you don't file this motion, you will be locked out, and then a month from now, the judge can find that you were right, and undo the default, but you will be living someplace else by then.