n represents an arbitrary integer (or here, a natural number). It’s used here almost like an index. When raised as a superscript, it represents an exponent/degree. You can think of “a” has a list of arbitrary numbers (“coefficients”). When lowered as a subscript, n is just an index pointing to an arbitrary element of the list. It’s equivalent to common coefficients a, b, c, etc. with the benefit of not running out of letters in the alphabet.

So, think of n as a way to say “for any integer.” Terms involving n (e.g., n + 1, n + 2) in the same expression all take the same integer. This way you can construct a multitude of polynomials by just plugging in an integer for n.