Originally Christians weren’t even called Christians. They were called "disciples" (i.e., "students") of Jesus of Nazareth. Later, in the city of Antioch, they received the name "Christians" (Acts 11:26). This probably happened in the A.D. 30s. This term spread very quickly—probably to the chagrin of those Jewish individuals who did not wish to acknowledge Jesus as the Messiah (Christ).
Ultimately, however, different groups began to break off from the Christian community, falling into either heresy or schism. These groups still wished to represent themselves as Christian—and many of them were, retaining valid baptism and a profession of faith in Christ. Consequently, a new word was needed to distinguish the Christians belonging to the Church that Christ founded from those belonging to the churches that had split off from it.
The term that was picked was kataholos, which means according to the whole or universal in Greek. The thought was apparently that these were Christians who believed and practiced according to what body of Christians as a whole did, in contrast to what some particular group thought or did. Over the course of time, kataholoscame to be represented by the parallel English word "Catholic."
Ignatius of Antioch did not introduce kataholos. However, his letters contain the earliest known uses of it. It may well have been used in other Christian writings prior to this, but we have simply lost them. It certainly was in general use in speech before this point, because Ignatius writes in such a way that he already expects his readers to know this term and what it means. He also uses the term in more than one of his letters, meaning that he expects people in more than one place to know the term.
This indicates that in his day—at the beginning of the second century (circa A.D. 107)—the term was already in widespread use. For it to be used in such a broad manner, it would have required some time to pass into currency in the Christian community, meaning that the term probably was coined sometime in the second half of the first century. We don’t know who first used it, but it was a suitable description of the Church Christ founded and so was already in general use by the time Ignatius wrote.