Is “summer” a common or proper noun and why?

It is a common noun. see explanation for details.

Proper are the nouns which describe unique things like cities: Paris, countries: Poland and so on.

Also names of people are proper nouns (like Peter) although there may be more than one people in the world sharing the same name.

In writing you can recognize proper nouns because they start with capital letter (except German language where all nouns are written with a capital letter no matter if they are proper or common)