What is the difference between the sets $A \times (B \times C)$ and $A \times B \times C$? How are the sets $A \times D$ (where $D = B \times C$ is the cartesian product of $B$ and $C$) and $A \times B \times C$ different? Aren't each element in both sets equal?
In particular, what is the difference between $\bigl(a, (b, c)\bigr)$ and $(a, b, c)$? If there is a difference, why is it significant?
 A: $A \times (B \times C)$ is a set of pairs, $A \times B \times C$ is a set of triples.
$A \times (B \times C) = \{(x,y) : x \in A, y \in B \times C\}$
$A \times B \times C = \{(a,b,c) : a \in A, b \in B, c \in C\}$
Of course, these sets are in natural bijection with each other, via the map $\phi : (a,b,c) \rightarrow (a,(b,c))$ and its inverse. Using the usual ZFC axioms, these are in fact isomorphic as sets.
A: The cartesian product is a binary operation that combines a pair of sets into a new set. It's only once you repeat the process with a third set that things could potentially get more interesting. The essential question is why is
$$
A \times (B \times C) \cong (A \times B) \times C 
$$
for any three sets $A$, $B$, and $C$. In other words, why is cartesian product associative?
It's only once we've established that they are "the same" in some meaningful sense that we can even write down a triple product like
$$
A \times B \times C 
$$
and have it be well-defined.
The two triple products are canonically isomorphic via the natural map
$$
\bigl(a, (b, c)\bigr) \mapsto \bigl((a, b), c\bigr). 
$$
First of all, this means that the map is an isomorphism of sets: it's bijective (one-to-one and onto) and hence has an inverse that is also bijective. But this isomorphism is canonical or natural, meaning it doesn't involve making any choices to construct the correspondence and is hence unique. (There's a deeper meaning from category theory that essentially characterizes this uniqueness in terms of how it interacts with other maps in and out of these product sets.)
What this means is that for all intents and purposes, you may consider the natural isomorphism as an equal sign and use either form for the triple product, including the neutral form whose elements are just triples of the form $(a, b, c)$.
Note: if you are interested in digging deeper into these subtle notions from category theory and you know a bit of linear algebra, read the answers here and the links within.
