Not a Roman, but here it is.
Papal infallibility means the Pope has divine protection on teachings of dogma and faith. Not everything he says is infallible. He is just guaranteed that the Church will have a certain harbour in times of doubt or trouble, specially in terms of doctrine and faith. For a statement to be infallible it has to be within that scope (doctrine or moral) and must be "ex cathedra", that is, "from the throne". That is not related to the physical throne, but to the more abstract concept of the "throne" being the dignity inherited from Peter, which includes stating in a clear way the faith believed by all Christians in all places and in all times. An ex cathedra statement of the Pope over matters of doctrine and/or moral will be infallibly right, even if the rest of the members of the Church disagree. That means that even if an entire generation adopts a heresy, the Church as whole will be protected in its sojourney on Earth by the promise of Christ to Peter through the Pope. Also it means the Pope may express with more clarity that which was just expressed in implicit ways before.
Romans must believe it to be in good stand with the Church. It is an intrisic aspect of their understanding of Ecclesiology, therefore on the very nature of the Body of Christ and how salvation is operated through it.
Most Romans I know though don't care about it, or think it is just an odd arrogant idea that history will sweep away any time. I too believe that sooner or later a generation of hierarchs will come in the Roman Church who will not think that the Church would be in trouble for being able to say "we got that wrong" even in matters of doctrine or faith. That will be a new dawn for Western Christianity, possibly for *all* Christianity, for Rome *was* given some proeminence throught Peter's and Paul's martyrdom and it's a sad thing they are hostages of such mistakes. I wouldn't be surprised to learn that many of the tears of the Theotokos we seen in miraculous icons, are for this, hopefully temporary, detachment of Rome from the rest of the Church.