Carl Lineaus invented the binomal (two name) system:
The Linennean system
Eukaryotes: cells have a nucleus - genes contain unused sections
Bacteria : calls have no nucleus - genes contain no unused sections
Archae: cells contain no nucleus - genes contain unused sections
Carl woese and others in the 1970's developed the idea of a higher level of classification than the kingdom called domain.
The evidence for Carl Woese's development came partly from the detailed analysis of the biochemistry of cell ribosomes and the way different cells reproduce.
Evolutionary trees:
The further up something is up the evolutionary tree, the more recently it branches from the other species on the tree. Species which are more closely related are closer on the tree.
DNA evidence is use to decide which species and organism belongs to. This can also be used to see how long ago different organisms had a similar ancestor, so how closely they were related.
For example, the red and giant panda both eat bamboo, both have a wrist thumb so it looks like they're related but based on their DNA, giant pandas are more closely related to bears and red pandas to racoons.
New evidence is used to reclassify organisms. Carl Woese separated bacteria from archaea. He found that archaea contain unused DNA sections, and have different cell walls. He created a third domain.
Classification is used to build Evolutionary trees, which show how related organisms are. Branches show the location of a common ancestor, which diverged into new species.