If R is a set of coefficients, and x is an indeterminant, i.e. a variable, the polynomials of x, with coefficients in R, are denoted R[x]. Note that R[x] becomes a new set of coefficients that might be applied to yet another indeterminant, such as y. Since the polynomials of y with coefficients in R[x] are the same as the polynomials in x and y with coefficients in R, we may use the notation R[x][y] or R[x,y].
Z[x] is the set of integer polynomials in x. Zm[x,y] is the set of polynomials in x and y whose coefficients are manipulated mod m.
Calling these items polynomials is merely a convenience, a familiarity. They are actually finite sequences of coefficients with rules for addition and multiplication. These rules happen to coincide with the rules for manipulating polynomials, so we call them polynomials. Actually the rules for adding and multiplying are almost inescapable, if we want the operators to be commutative, associative, and distributive. Sometimes xy is different from yx, and on rare occasions 3x is different from x3, but most of the time R[x,y] means the standard polynomials in x and y with coefficients in R.
A power series is a polynomial that goes on forever. The set of power series with coefficients in R is denoted R[[x]]. The standard polynomial rules apply for addition and multiplication. Note that R[[x]][[y]] is the same as R[[x,y]].
The laurent series with coefficients in R is denoted R((x)). This is a power series running in both directions. The indeterminant x can have positive or negative exponents. However, there are finitely many terms with negative exponents. The standard polynomial rules apply for addition and multiplication.
If F is a field, then F(x) is a field extension, the smallest field that contains F and x. If x is some algebraic element over F, then F(x) is a finite extension. If x is an indeterminant then F(x) is all rational functions over F. (A rational function is the quotient of two polynomials.) Why then do we use parentheses for F((x))? Because the laurent series over F form a field. Use synthetic division forever to divide one series by another. By dividing two polynomials, F((x)) contains F(x). However, the former may be larger. If F is finite, F(x) is countable, while F((x)) is uncountable.