I previously discussed injunctive relief and the basics of an appeal. In appealing a district court judgment, a party can move to stay or for an injunction of the judgment until the appeal is completed. In this context, a stay of the judgment is effectively the same as an injunction. Both prevent the winning party from enforcing the district court judgment until the appeal is over.
A motion to stay or for an injunction of the judgment is generally filed in district court. Such a motion is only filed with the court of appeals if it is impracticable to file it in district court, or if the motion has already been denied in district court. If a party files neither a motion to stay nor for an injunction, the district court judgment is effective on appeal.
In deciding whether to grant a stay or injunction pending appeal, the court's considerations include the likelihood of success on appeal. If the same judge who presided over the case in district court rules on the motion (which is likely), he or she will not likely consider the likelihood of success on appeal to be very high. This weighs in favor of filing the motion in the court of appeals. However, if you file the motion in district court first, you get two chances for the motion to be granted. This weighs in favor of filing the motion in district court.