Diversity means variety and biodiversity refers to variety of living things.
Existence\ occurrence of varied form of living organisms [plants, animals, microbes] in a particular area is known as biodiversity.
These variations are in terms of physical, physiological, psychological, behavioral characteristics among living organisms from all sources including terrestrial, aquatic and other ecosystems. Like: Size\ color\ shape\ height\ food habit\ age etc.
Classification is defined as the systematic arrangement of living organisms into groups and sub-groups on the basis of their similarities and dissimilarities.
Organisms having similar characteristic features are placed in same group and those having dissimilar features are placed in different group. Also when dissimilarities appear in same group they are further sub-divided.
The branch of science that deals with the classification of organisms is known as taxonomy.