This is one of those spiritual gifts that doesn't get a lot of attention, probably because it's not really clear what it means. Here's Joseph Fielding McConkie and Craig Ostler's take from Revelations of the Restoration, p. 366:
The diversity of gifts found in the community of Saints suggests that there are a host of tasks that need doing and a variety of ways in which they can be done. It is expected that all faithful Saints will labor in their callings, using the gifts that God has given them. For this reason people are called to serve in various capacities for a time when their gift or gifts are particularly needed. We find the apostle Paul saying, "I have planted, Apollos watered" (1 Corinthians 3:6), to which it might be added, another pruned, and still another harvested, each doing that for which he was best suited. Surely, this has been true with those men called to stand at the head of the Church. EAch has been endowed with the particular gifts needed during the time of his administration. The Prophet Joseph laid the foundation of this dispensation as no other man could. Yet, perhaps he would not have been Brigham's match in leading the Saints across the plains and colonizing the Great Basin. And so it has been with one prophet after another, each specially prepared to meet the challenges of that time for which he was called.
The "diversity of operations" could also be understood to embrace different ways particular gifts are manifest. The gift of healing, for instance, finds expression in one person's ability to administer healing herbs, the ability of another to heal through a priesthood blessing, and in still another the power to say, "Rise up and walk" (Acts 3:6). Indeed, we read of one who was healed simply by touching the hem of the Master's garment (Matthew 9:20-23).