Is "summer" a common or proper noun and why?
It is a common noun. See explanation for details.
Proper nouns are the nouns which describe unique things like locations: Paris (a city), Poland (a country), Asia (a continent), and so on. They are always capitalized.
Names of people are also proper nouns (like Peter) although there may be more than one person in the world sharing the same name.
In writing you can recognize proper nouns because they start with capital letter (except in German where all nouns are written with a capital letter no matter if they are proper or common)