Taxonomy is the science or organizing things by giving things names then ranking them in a hierarchy (ie, Class --> Order --> family -> Genus -> Species). This description of all living things in this manner predates Linneaus, and his method if binomial scientific names (Genus --> Species) completed the hierarchy and streamlines each species name into two names. Previous to this some organisms has only a single name (Example: "Rosa for the garden rose while others had more than one name (Example:).

Until the last half of the 20th century, all this was done by differentiation of morphological characters - in short, things that looked alike were put together as groups. Later the bones and scales of the head became another method as this better reflected evolution and by 1981 Parenti had used this to sort out family relationships in the killifish.

Later the science of molecular biology came to the rescue with genetic analysis techniques wherein the DNA was examined - not as a whole, but just parts of it. Many parts to be sure, but still a fraction of the whole genome.

The good news is it is a very useful tool that can sort out family relationships which in the majority of cases is congruent to established taxonomy based on morphological characters and there have indeed been a few surprises where things turn out to be an a radically different family or genus than what morphological characters would have suggested otherwise.

Keep in mind this is all because of evolution. Think about it, the Linnean taxonomy system predates Darwin and while there where theories prior ro Darwin that suggested things turned into other things it was little more than a fringe theory.

The advent of beginning to understand evolution combined with the cracking of the genetic code at a molecular level (to say nothing of the Internet) means we can look at things another say now, rather than lump all red things together or something equally silly (Don't laugh this has been done)