In the Creed, we confess our faith in "one holy catholic and apostolic church". This phrase is a stumbling block for some, because when they hear the word 'catholic' they immediately think Roman Catholic, and of course they don't believe in the Roman Catholic Church in that sense. For that reason, the word 'catholic' is sometimes dropped and replaced with 'universal'. I have no particular problem with that switch, although I would on the whole prefer to retain the word 'catholic' and explain its meaning. It is true that in the Roman Church, the word 'catholic' is thought to refer to the universal validity of that church which is in communion with the Pope, with its clerical hierarchy and congregations. To be outside the Roman communion is to be (to some extent; the line has become a little more fuzzy for post-Vatican 2 Roman Catholicism) outside the catholic church. But I don't think we must, or should, accept the Roman construal of catholicism. Let me try to offer an alternative.
To believe in the catholic church is to believe that Christ has but one people, one body. This is the church. It is one across the centuries, and it is one throughout the world. It's unity is not direct, but indirect; by which I mean, the members are not joined directly to one another, but are all joined in the one Holy Spirit to Christ. The catholicism of the church, therefore, does not rest on any human organisational scheme, whether that of Rome or anything else; it rests in a common faith in the Lord Jesus. This commonality may well be only imperfectly expressed, or even sometimes completely hidden, in this world, but since it is grounded in Christ it cannot be ultimately broken and will be ultimately revealed.
That's how I understand catholicism. But to get at what it means in practice, it is perhaps more useful to ask what a catholic spirit looks like, and to illustrate that by contrasting it with its opposite, sectarianism. In essence, the catholic spirit draws the boundaries of the church as broadly as possible, where the sectarian spirit tends toward narrowness. There are lots of ways in which this plays out.
The catholic sees an essential unity between the church of the past and the church of the present, and looks on the theological and creedal decisions of the past as having (relative) authority within the church. The sectarian, by contrast, is free to reject the past, and tends to be disparaging of the church in past ages.
The catholic sees their own church as part of a greater whole, and is therefore free to draw upon liturgical and theological resources from around the world, throughout time, and across a broad ecclesial spectrum. The sectarian tends to make use only of resources from their own particular tradition, or in more extreme cases only things tailor-made for their own congregation and situation.
Again, the catholic sees their own church as part of a greater whole, and therefore wants to bring the particular insights and strengths of their tradition to the rest of the church in service. The sectarian is happy just doing their own thing.
The catholic can't be content with the divided nature of the church, but seeks a clearer expression of the essential unity of the church. This will involve entering into controversy - the catholic is not content to see parts of the church affected by theological error. The sectarian, on the other hand, either adopts a 'live and let live' attitude to churches of different traditions (i.e., indifference), or writes off any church which significantly disagrees with his own position as outside the church altogether.
Examples could be multiplied, but you get the idea.
Be more catholic.