how do classification systems help us to think about and study diversity on earth?

Taxonomic classification systems are useful for a couple of significant reasons. First and foremost, it creates a name for each living organism that ALL scientists can agree upon. Therefore, biologists on opposite sides of the world can be confident that they are discussing the same organism when looking at the taxonomic label of that organism. Secondly, the design of the system is useful for showing evolutionary relationships among organisms. Generically, the breakdown is kingdom, phylum, class, order, family, genus and species. There are five kingdoms (six in some systems) and millions upon millions of species. For any two organisms, they are more closely related evolutionarily if they share a classification more toward the species. For example, two organisms that share a genus are more closely related than two other organisms that only share a family. You should know that there modifications of the above system so that you get things like subphyla, superfamilies, etc. The genus and species names are GENeral and SPECIfic for a particular organism and are always written in italics, with the genus name being capitalized and the species name beginning with a lower-case letter. The millions of species described by the system is a testimony to the great biodiversity of life on this planet.

Listen.... and listen good:

If you want a detailed answer then you ought to put more detail into explaining your question.

For instance:

#1. Classification of what? The number of things in this Universe or even just on this planet that can be classified is absolutely cosmic.

#2. Diversity of what? I assume you're thinking of some sort of taxonomic unit, but there are way too many out there for us to know what you're talking about. If you're discussing biodiversity in general... i.e. all life.... then please specify...

Please addend to your question, and I may.... MAY ... try to answer.

Rest assured, I know. I have a Palaeobiology degree, and biodiversity is a big part of that.