What are the details behind structures such as $F[X]$? I see the notation $F[X]$ often, where $F$ is an algebraic structure (usually a field). The notation $\Bbb R[X]$ has never been explained to us in class, other than that it refers to the polynomials for some reason.
I think that $F[X]$ refers to the set generated (through multiplication and addition) by $F \cup \{X\}$, is this correct?
If so, I also once saw the notation $\Bbb Z[X_1,...X_n][X]$. How is this any different from $\Bbb Z[X_1,...X_{n+1}]$?
 A: Not an answer, but a clarification of some confusing notation. Needless to say, most people will consider the below to be a long-winded nitpick, but this is one of the somewhat confusing points of basic algebra, and it wouldn't harm to lessen the confusion.
There are at least two different meanings of the notation $F\left[X\right]$ when $F$ is a commutative ring and $X$ is some symbol.
The first meaning of $F\left[X\right]$ is "the polynomial ring over $F$ in an indeterminate which is called $X$"; this is how the notation $F\left[X\right]$ is commonly understood when $X$ is just a symbol which has not been introduced so far. Note that this makes sense even if $F$ is not commutative; in this case, the standard definition of the polynomial ring still applies (the indeterminate $X$ then commutes with all elements of $F\left[X\right]$).
The second meaning of $F\left[X\right]$ is "the $F$-subalgebra of $B$ generated by $X$, where $B$ is whatever $F$-algebra $X$ lies in". This, of course, only makes sense if $X$ is an already-defined object rather than a new symbol.
Of course, once one has defined a polynomial ring $F\left[X\right]$ in an indeterminate $X$ over $F$, the symbol $X$ is no longer new but denotes a particular element of this polynomial ring $F\left[X\right]$. So, after $F\left[X\right]$ is defined, $F\left[X\right]$ can be understood in both of the above two meanings. Fortunately, these meanings boil down to the same thing in this case, because the polynomial ring $F\left[X\right]$ is the $F$-subalgebra of $F\left[X\right]$ generated by $X$ as well. (If this wasn't the case, the notation $F\left[X\right]$ would be horrible.)
(Occasionally, when $X$ is a set or a family, $F\left[X\right]$ means something slightly different from what I have defined above; but it is very similar: basically, instead of $X$, the elements of $X$ are being used as indeterminates or generators.)
While $\mathbb Z\left[X_1,X_2,...,X_n\right]\left[X_{n+1}\right]$ and $\mathbb Z\left[X_1,X_2,...,X_{n+1}\right]$ are an example of two isomorphic rings, it is important to keep them apart for the sake of sane notation. If $P$ is a polynomial in $\mathbb Z\left[X_1,X_2,...,X_n\right]\left[X_{n+1}\right]$, then the constant term of $P$ is the part of $P$ where $X_{n+1}$ does not appear; this is an element of $\mathbb Z\left[X_1,X_2,...,X_n\right]$. If $P$ is a polynomial in $\mathbb Z\left[X_1,X_2,...,X_{n+1}\right]$, then the constant term of $P$ is the part of $P$ where no variables at all appear; this is just an integer. So if you identify $\mathbb Z\left[X_1,X_2,...,X_n\right]\left[X_{n+1}\right]$ with $\mathbb Z\left[X_1,X_2,...,X_{n+1}\right]$ by abuse of notation, you are making notions like "coefficient", "constant term" etc. ambiguous. (This is OK if you explicitly dissolve the ambiguity.)
A: Let us suppose $F$ is a field. You may think of the set $F[X]$ in at least three ways (but only the third one is a precise answer to your question):


*

*Polynomials: the generic element of $F[X]$ has the form $p(X)=\sum_{i=0}^na_iX^i$,
where $a_i\in F$ are called the "coefficients" of the polynomial, and $n<\infty$.

*Infinite "rows" of elements of $F$, something of the form $(a_0,a_1,a_2,\dots)$, with almost all $a_i=0$.

*The more precise (and general) definition is as follows: for any $n\geq 1$,
$$F[X_1,\dots,X_n]=\bigoplus_{i\geq 0}\textrm{Sym}^i\,V_n,$$
where $V_n=\textrm{Hom}_F(F^n,F)$ is the vector space of linear forms on $F^n$.


You have ring isomorphisms between the three structures here defined (once you know what these structure are of course). What really happens in passing from $F$ to $F[X]$ is that you "add a formal variable" $X$, a symbol that might live very far from $F$! This is somewhat delicate and it is not explicitly specified in case 1, totally bypassed in case 2, but (hopefully) made clear in 3.
The fact of adding this formal variable, or indeterminate, makes the resulting ring $F[X]$ an infinite dimensional vector space over $F$.
By the way, in each case it is easy to see that $F[X_1,\dots,X_n][X_{n+1}]\cong F[X_1,\dots,X_{n+1}]$.
A: For the second question, 
$\mathbb{Z}[X_1,\ldots,X_n][X] = (\mathbb{Z}[X_1,\ldots,X_n])[X]$ is the ring of polynomials in $X$ with coefficients in $\mathbb{Z}[X_1,\ldots,X_n]$ (which is the ring of $n$-variable polynomials over $\mathbb{Z}$). But as you suggest, there is an (almost obvious) isomorphism
$$\mathbb{Z}[X_1,\ldots,X_n][X] \cong \mathbb{Z}[X_1,\ldots,X_n,X] $$
A: This is just an addition to the answer of @atricolf:
There is another way to see the polynomial ring over an arbitary ring, which comes from the more general notion of a monoid ring (you can check that on wikipedia).
The polynomial ring $R[X]$ is just the monoid ring $R[(\mathbb{N_0},+)]$. I think this is the nicest way to formally introduce polynomial rings.
Basically, this is option 2 of atricolf, but from a more general perspective. In general the concept of a monoid ring serves well to model things like "linear combinations of some thing with coefficients in some other thing" on a formal viewpoint.
