$\bigcap\limits_{n=1}^\infty A_n$ vs $\bigcap\limits_\infty^{n = 1} A_n$ This is probably a trivial question, but what is the difference between those two?
$$\bigcap_{n=1}^\infty = \{x \mid \forall  n \in \mathbb N, x \in A_n\}$$
What does the other intersection mean?

Usually we define merely
$$\bigcap_{i\in I}A_i=\{x\mid \forall i\in I\colon x\in A_i\}$$
where $I$ is some nonempty index set.
Alternate notations are common for two special cases: 
(i) If $I=\{n, n+1, \ldots, m\}$, we write
$\bigcup_{i=n}^m A_i$ for $\bigcup_{i\in I}A_i$.
(ii) If $I=\{i\in\mathbb N\mid i\ge n\}$, we write $\bigcup_{i=n}^\infty A_i$ for $\bigcup_{i\in I}A_i$.
The notation you exhibit has never occured to me. It doesn't match notations for e.g. sums with $\Sigma$ either.
