The courts are required to judge cases concerning injuries caused by a person’s animals, whether they injure humans (as in this case) or property (as in verse 35, wherein one person’s ox gores another person’s ox). The damage is not limited to goring, as in the Torah’s examples. Rather, it includes biting, trampling and general clumsiness. The Torah uses the term goring to describe the way an ox might kill a person because that is the way such a thing might normally occur, but it is not meant to be exclusive. (Cross reference this mitzvah with #55.)
When judging a case, the courts must consider various factors such as whether or not the animal had a history of violence. (In such a case, its owner should have known to take extra precautions.) Also relevant are the type of animal, the venue where the damage occurred and other factors that mitigate the degree of liability.
This mitzvah applies to the courts, not to individuals. The specific fines and penalties described by the Torah can only be imposed by duly-ordained judges in Temple times, but that doesn’t mean that in other times one would be exempt for paying for the damage done by one’s animals.
The obligation to judge such cases is discussed in the first six chapters of the Talmudic tractate of Baba Kama. In the Shulchan Aruch, it’s found in Choshen Mishpat 389. It is #237 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos.