What is the basis for classification?

The basis of classification depends on what is to be classified and what is the purpose of classification. In general, a set of characteristics (one or more characteristic criterion) are used for classification of any object, generally, for the purpose of comparison. For example, we can classify a group of 12-15 year old students into different classes based on body weight (slim, average, overweight, etc.), height (short, average, tall, etc.), etc. Such a classification will help us compare them. We can classify elements into metal and non-metals, reactive and inert, natural and synthetic, etc. Similarly, we can classify animals into vertebrates and invertebrates (depending on presence or absence of backbone), macroorganism and microorganism (depending on size), prokaryotes and eukaryotes (depending on presence of one cell or multiple cells), etc.  

In each of these example cases, something has been classified on the basis of a characteristic and for comparison purpose.

Hope this helps.