On my algebra course, sometimes we write, say
$$f \in R[X], f= X^2 + X + 1$$
And sometimes we treat polynomials as functions, so
$$ f(x) = x^2 + x + 1$$
What is the difference between these two ways of writing a polynomial?
I'm a bit confused about the two different notations. Is it that if $f = X^2 + X + 1$, then $f(x) = x^2 + x + 1$?
It depends on what you want to do. Sometimes you want to treat polynomials as elements of the polynomial ring, and sometimes you want to treat polynomials as functions. Conflating the two is a common form of abuse of notation, or what in computer science is called overloading.
In this case there is actually a precise way to describe the relationship between these two descriptions of polynomials: if $R$ is a commutative ring, then the set of homomorphisms of $R$-algebras from $R[x]$ to $R[x]$ can be canonically identified with $R[x]$ because $R[x]$ is the free $R$-algebra on one generator; moreover, composition of homomorphisms corresponds to composition of polynomials. There is an even better way to say what this means using category theory but maybe that's going too far for now.