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Taxonomy is a hierarchical system of classifying and naming things. The taxonomy currently in use for biological organisms, the Linnaean taxonomy, is more than 250 years old and even predates the theory of evolution. Carolus Linnaeus, a Swedish botanist, developed the system to organize living things by morphology - the physical characteristics that they have in common.
As we now know, organisms that share physical characteristics aren't necessarily related to each other. This is a problem for the Linnaean system, and it happened because scientists of that time didn't have any way to know which organisms were actually related. Fortunately, now they do. The Linnaean taxonomy has been and continues to be updated, shifting more towards genetic relationships rather than physical similarities. Right now it's still a mixed bag. The lesson here is that while scientific names are useful as a common language, they are not biological facts.
A new competing taxonomy called PhyloCode attempts to fix the classification problems by starting over with an entirely new hierarchy. This means that every living thing would get a new scientific name, although many of those names would probably resemble the Linnaean species names. I won't even hazard a guess as to whether or not PhyloCode will ever catch on. Compared to the Linnaean system, PhyloCode is still in its infancy.
The felid (cat) taxon (under the Linnaean system) has changed quite a bit over the years. For example, the smaller cats all previously had a genus of Felis, but by now most of those cats have new names, and you will find both old and new names in the literature. Below is the current felid taxon, but I expect it will change eventually. The extra column called Genetic Lineage is not part of the taxon. I've included it to show that relationships based on DNA comparison hint at a somewhat different taxonomy.
Subfamily Acinonychinae
Subfamily Felinae
Subfamily Pantherine