Readers may notice from my profile that I refer to myself as a "Traditional Catholic." They may also be aware of the various groups and organizations of traditional Catholics that exist in the Church today - such as the Society of St. Pius X, the Fraternity of Saint Peter, and the Institute of Christ the King, etc. What is different about traditionalists? Isn't it enough to just be Catholic, and not have to label and distinguish some Catholics from others? Unfortunately, the answer is no, because there is a distinction between Catholics, there is difference among them. It shouldn't be so, because, I think, to be Catholic in the fullest sense just means to be traditional. Hence I think it is unfortunate that we should have to label some Catholics as traditional and others as not; but it must be done.
Anyway, I have been puzzling for some time now over the problem of what exactly sets traditionalists like myself apart from other Catholics. There is definitely a difference. But the more I've studied the problem, the more have I come to realize that it isn't very simple or cut-and-dry, and that I can't really give a simple and concise explanation. In a summary blog page like this, I can only point out some general characteristics; but there will still remain much to be said, addressing particular problems relevant to traditional Catholicism, which will have to be dealt with elsewhere. So I've decided to try to just point out some general things here, and be satisfied with that, for now.
This is one of those huge questions that can't be dealt with in a summary post like this, so I'll have to try and tackle it - learning about it, myself, in the process - in other posts. But in general, I think it would be right to say that traditionalists have a very great respect for Catholic Tradition in its many forms. There is the Divine Tradition, intrinsic to the Deposit of Faith, a means of transmitting Divine Revelation itself, and therefore unchangeable. There are the traditional teachings of the Church which, though not part of the Deposit, nonetheless constitute an important part of her teaching heritage - some are changeable, some are not. There are the many other lesser, but still very important, traditions of theology and spirituality, passed down to us by the saints, fathers, doctors, and theologians of the Church, constituting another very large and important part of her heritage and patrimony (in particular, the theology of St. Thomas and the scholastics is immensely important to the Church). There are also the traditional customs, practices, and disciplines in the Church by which the faithful have lived and practiced their faith - Lenten practices, for example; or the abstinence from meat on all Fridays; and many other such things. And of course, very importantly, there is the Traditional liturgy of the Church, especially the Traditional Latin Mass, also called the Tridentine mass, or the Extraordinary Form (I will speak about this below). Etc. Now all this isn't a very precise or systematic way of speaking about the different types of tradition; again, that will have to be done elsewhere, and I think I myself still have to learn a lot more about it first. However, I think this little summary gives a good enough idea of what it is that identifies traditionalists as such. Basically, they have a very great respect for all of these traditions, adhering strictly to the Divine Tradition and striving to incorporate the smaller traditions into their lives as Catholics.
2. Change and Novelty. Another defining general characteristic of Traditional Catholicism is its attitude towards change. Now, it ought to be recognized that certain parts of Catholic Tradition cannot change, while other parts can indeed change. In particular, the Divine Tradition, which is intrinsic to the Deposit of Faith, cannot change; and there are other teachings which, though not intrinsic to the Deposit, must nevertheless be unchanged in order that the truths of the Deposit itself be preserved. Other parts of the tradition are changeable. However, for the traditionalist, whenever change does occur, it ought only to occur when it would really be conducive to the common good of the faithful. This is in accordance with the teaching of St. Thomas Aquinas on the changeability of human laws (I-II, q.97-a.2) Thus, traditionalists acknowledge that there can be good and legitimate developments within tradition. Otherwise, change is of itself most likely to be quite detrimental, rather than beneficial. Furthermore, quite rarely should it be brought about suddenly, but should have rather the character of gradual, organic development. Novelty is disruptive; but organic growth and development is healthy. This is how traditionalists look at everything they have received from the Church: they seek to adhere, in general, to the traditions, guarding against novelty, admitting only gradual changes and developments when they are really beneficial. Whereas today there is an overly strong tendency to change what has been passed down to us. There is an almost obsessive fascination for change, as if just for its own sake, and a desire to "adapt" the Church to the ever evolving life and history of man. But traditionalists hold this to be a very serious mistake. The Church is stable before she is changeable; she is first the Church of God, Who is immutable, and secondly the Church of man, who can change. Her divine element is primary to her human element. Hence it is a mistake to hold as a matter of principle that the Church must adapt to the times, to the changes in human life and history; and this seems to be very much the mindset of mainstream Catholicism today. It is an effect of the heresy of Modernism, to which traditionalists are very staunchly opposed (and which, again, will have to be dealt with later, as it is a huge topic...). And not only this, but because the traditions are received from the Church's many saints and theologians, it is highly unlikely that change could be introduced without it being in some way inferior to what the saints have passed on; thus, aside from being undesirable just as change, in practice it is also highly unlikely to be more beneficial, as we can hardly presume to know better than our ancestors and the saints.
I mentioned this earlier, but I think it definitely deserves more attention for itself. This is because I think the one characteristic which stands out the most is the Tridentine Mass, or the Extraordinary Form. When other Catholics think of traditional Catholics, then immediately the very first thing that will jump to their minds is the Tridentine Mass. Traditionalists favor the Tridentine over the Novus ordo missae for several reasons. First, I should note that I do not approach the extreme of some traditionalists who deny that the Novus ordo is even a valid mass; on the contrary I hold it to be perfectly valid, promulgated by the disciplinary infallibility of the Church (big topic). Thus, I do not believe that it contains any doctrinal errors within itself. Nonetheless, I think it contains many weaknesses, both in the way it is typically celebrated and even in the text itself. Traditionalists oppose the practices of the priest facing the people, having mass completely in the vernacular, receiving communion in the hand, extraordinary ministers, lay readers, altar girls, general feminization of the liturgy, secular styles of music, liturgical dancing, clapping, holding hands, embracing at the kiss of peace, the endless noise and activity, and so on and so forth. These things constitute a break from tradition. Now although it can be argued that these are accidental to the the Novus ordo form of the mass itself, nonetheless I think it must be admitted that the Novus ordo seems to open the door for all these things, because of the fact that it allows for much personal interpretation of the liturgy. What's more, the text of the mass itself contains many ambiguities and omits many important things relating to doctrine, thereby allowing for wrong theological interpretations, potentially leading minds away from the truth, and from the traditional understanding of Church teaching, etc. And quite unsurprisingly, this is all the product of the Modernist mentality of change which I just described in the last paragraph. The Novus ordo is a fabrication, a novelty, a kind of break from tradition, and not an organic development of the mass. But anyway, it would require a separate post, or several of them, for me to explain all of these problems in more detail. In general, this is what traditionalists believe about the Novus ordo Mass. And so they prefer to stick with the Tridentine Mass, in which the text is much more doctrinally solid; the music consists of Gregorian chant, polyphony, and the organ; all the servers are male; there are many periods of silence; the mass is said completely in Latin (minus the Kyrie, in Greek); there is less room for personalization of the liturgy; and so on. This is the mass of the ages, the mass of tradition, the heritage of all Catholics.
There is much more that could be said, regarding all of the above, and many other topics as well, all relevant to traditional Catholicism. There are all the problems of Modernism and neo-Modernism, the New Theology, Vatican II, religious liberty, ecumenism, salvation outside the Church, ecclesiology, and so forth. I hope to be able to address more of these subjects here on the blog, eventually.