You won’t find any Phoenix Downs or Max Revives in this game! Instead, reviving characters requires you to sacrifice a bar of your Party Gauge. The Party Gauge can be filled by using moves correctly (such as using Back Slash from behind) and by combining certain Arts. It may be wise to avoid using a Chain Attack in a difficult battle, as you may need to revive a party member instead.
A party member who loses all of their HP during battle will be automatically revived at the end of it. There are also Gems in Xenoblade that increase the amount of HP recovered when you revive a party member. Some Auras will grant HP just before a party member is knocked out or incapacitated.
In the image, nobody needs to be revived, but Reyn has low HP indicated by the red highlight on his picture. In the upperleft corner is the Party Gauge used for a variety of things, including reviving characters. The party has three bars which means party members could be revived 3 times, if not used for something else.