The second part of the word comes from the Greek root hol- which is manifested in words like holistic and hologram. The word 'catholic' means "with respect to everything" or "concerning everything" or "all-embracing" or "universal" - and we see this in nonreligious expressions like "he is a man of catholic tastes" which simply means that he has eclectic tastes.
So, then, when the church claims to be "catholic," it is merely expressing that it is a collection of all those who follow Jesus, regardless of where they live, which language they speak, or which culture informs their social context. That we have a "catholic" faith is simply saying that God loves all humans, that Jesus offers forgiveness to all humans, and that eternal life in heaven is made available to all humans.
The confusions begin to emerge because one particular institution is called the "Roman Catholic Church" - a paradox created by the words alone. The word "Roman" refers to a concrete specific place, and connotes certain traditional social structures which accompany it, while the word "catholic" denotes just the opposite, something which transcends a particular physical situation.
The phrase 'church universal' or 'invisible church' is often used to refer to the collection of all people who are followers of Jesus, regardless of which, if any, specific 'church' or denomination they embrace. The word 'church' generates ambiguity, as does the word 'catholic' - and the two of them together create a downright conceptual jungle.
Although we use words to attempt to clarify and articulate our concepts, they often risk doing just the opposite. It is at least a step toward clarity to distinguish the "Roman Catholic" church, which is a specific institution, from the invisible universal "catholic" church, which is a set of people with no regard to membership in a particular organization or denomination.
Mark Dever offers a historical insight into the development of this terminology. Going back in history, before the phrase "Roman Catholic" emerged, which it did after Luther's Reformation in the 1500s, we can see the word 'catholic' being used when there was less confusion or ambiguity surrounding it:
As far as we know, Ignatius of Antioch was the first person to use the word catholic in relation to the church. In his letter to the Smyrnaeans, written around A.D. 112, he wrote, "Where Jesus Christ is, there is the universal church." Early Christian writers believed in the catholic church — that Christians everywhere trusted in one God, confessed one faith, received one baptism, and shared one mission. In that sense, catholic meant "real" or "authentic."
The earliest institutional churches were geographically defined: the Chaldean Church, the Ethiopic Church, the Syrian Church, etc., and so the Roman Church was simply institutional Christianity as it occurred approximately within the boundaries of what was, or had been, the Roman Empire.
There were different varieties of Christianity, but each was a sort of monopoly within its geographic area. Only after the Reformation did a situation arise in which different variations of Christianity existed side-by-side within the same region. Despite the notion of "religious wars," they coexisted peacefully for the most part, and when so-called "religious wars" arose, they were often motivated by quite nonreligious political ambition, poorly disguised by a thin veneer of religious vocabulary.
A new vocabulary was needed for this situation, a situation of various flavors of Christianity heterogeneously occupying the same space. Thus arose a colorful spiritual landscape of Lutherans, Calvinists, Anglicans, and Roman Catholics. Because 'catholic' is already three syllables, and 'Roman Catholic' is five, 'catholic' quickly became shorthand for 'Roman Catholic' - generating confusion. Generations of Lutherans, Anglicans, and Calvinists have stated that they belong to the catholic church, but not to the Roman Catholic church.
In addition to meaning 'authentic', the word also came to denote 'orthodox' - or, simply put, 'true' or 'accurate' - in contrast to institutions which might promote specious or incorrect ideas. As we saw, above, from Ignatius of Antioch, who died around 110 A.D., his notion of the church's catholicity meant that it was actually existing as a thing or occurring in fact, and not something merely not imagined or supposed, and that it was genuine, and not an imitation or something artificial.
Clement of Alexandria, who was born around 150 A.D., exemplifies this added layer of meaning - orthodoxy on top of authenticity - in a passage cited by Mark Dever:
There is one true Church, the really ancient Church, into which are enrolled those who are righteous according to God's ordinance ... The one Church is violently split up by the heretics into many sects. In essence, in idea, in origin, in pre-eminence we say that the ancient Catholic Church is the only church. This Church brings together, by the will of the one God through the one Lord, into the unity of the one faith which is according to the respective covenants (or rather according to the one covenant established at various times), those who were already appointed; whom God fore-ordained, knowing before the world's foundation that they would be righteous. The preeminence of the church, just as the origin of its constitution, depends on its absolute unity: it excels all other things, and had no equal or rival.
Another translation of Clement's thorny syntax confirms his perception of the tension between the one and the many: centuries before the Great Schism of 1054 A.D., and centuries before Luther's Reformation, there were divisions of other kinds, and while those divisions may have been quite real in the concrete functioning of institutions, and in the specific points of doctrinal statement, there is still a notion of the church's oneness:
From what has been said, then, it is my opinion that the true Church, that which is really ancient, is one, and that in it those who according to God’s purpose are just, are enrolled. For from the very reason that God is one, and the Lord one, that which is in the highest degree honorable is lauded in consequence of its singleness, being an imitation of the one first principle. In the nature of the One, then, is associated in a joint heritage the one Church, which they strive to cut asunder into many sects. Therefore in substance and idea, in origin, in pre-eminence, we say that the ancient and Catholic Church is alone, collecting as it does into the unity of the one faith — which results from the peculiar Testaments, or rather the one Testament in different times by the will of the one God, through one Lord — those already ordained, whom God predestinated, knowing before the foundation of the world that they would be righteous. But the pre-eminence of the Church, as the principle of union, is, in its oneness, in this surpassing all things else, and having nothing like or equal to itself.
Around 350 A.D., Cyril of Jerusalem nudged the word 'catholic' a little closer to its modern meaning, while yet retaining its previous meanings. He spoke of geographical diversity and social class diversity; he spoke of a church which is universal because it addresses those things which are precisely human, those things which all humans face or need. Mark Dever cites this passage:
It is called Catholic then because it extends over all the world, from one end of the earth to the other; and because it teaches universally and completely one and all the doctrines which ought to come to men’s knowledge, concerning things both visible and invisible, heavenly and earthly; and because it brings into subjection to godliness the whole race of mankind, governors and governed, learned and unlearned; and because it universally treats and heals the whole class of sins, which are committed by soul or body, and possesses in itself every form of virtue which is named, both in deeds and words, and in every kind of spiritual gifts.
As the followers of Jesus are called to clarify their task again to each new generation - semper reformanda - Mark Dever sees four challenges to clarifying the catholicity of the church in the twenty-first century. The church must overcome, first, provincialism: we are inclined to project our social, cultural, and geographic specifics onto the universal church, rather than seeing the universal truth of Jesus weaving its way through various traditions, cultures, and places; second, sectarianism: which is not to minimize or deny the significance of doctrinal and dogmatic differences, but rather to see what might be maintained and achieved in spite of them, as we work together on those tasks which God has assigned to us; third, racism: while much has been improved concerning this topic, much remains to be done, and the challenge grows as immigrants and emigrants flow through the continents of Asia, Africa, Europe, and the Americas; fourth, exclusivism: the Good News about Jesus is for all people, and no socio-demographic variable should exclude anyone from fellowship - marital status, income level, profession, etc.
A quick review of John's Gospel, chapter seventeen, or Paul's letter to the Ephesians, chapter four, will suffice as evidence from Scripture, despite the fact that the word 'catholic' or its Greek equivalent do not appear in the New Testament.
While millions of Lutherans, Anglicans, and Calvinists are not members of the Roman Catholic church, they are members of the universal invisible catholic church - as are millions of Eastern Orthodox (Russian, Greek, etc.), millions of Copts, millions who belong to the Chaldean Church or to the Syrian Church or to the Ethiopic Church. And, of course, equally, members of the Roman Catholic church are also reminded to remember that they are part of this invisible universal church.