Specialties are skills that give your party advantages not necessarily associated with battle. For the example, some allow you to acquire certain items, while other give you more information on the minimap. They can be unlocked by various methods like completing sidequests and performing Private Actions. Most of the very helpful specialties are acquired by completing Welch's sidequests so make sure to visit her often in her lab in Myiddok!
You can view your party’s specialties and level them up in the Specialties menu under the Skills section of the main menu. Although most specialties are passive and take effect immediately upon learning them, there are specialties that must be activated manually. For these, you have to press Square button while in the aforementioned menu to use them.