What does this double sided arrow $\longleftrightarrow$ mean?

In the area of logic, $\longleftrightarrow$ is usually used for "if and only if" instead of $\iff$ (because who wants to bother drawing that second line all the time).

Otherwise when dealing with functions, $\longleftrightarrow$ might also be used to denote a bijective function. So $f \colon A \leftrightarrow B$ is a bijection between $A$ and $B$. Or you could similarly write $$ A \overset{f}{\longleftrightarrow} B $$

In regards to what was likely meant in the video that you saw, the following is true:

For a given value of $x$, one has $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges if and only if $\int\limits_{1}^\infty \frac{1}{t^x}dt$ converges.

On the one hand, $\longleftrightarrow $ is used for connecting propositional formulas (e.g. $p\to q \lor (p\longleftrightarrow q) \land \lnot w$). You can understand it as a binary operator like AND or OR, which are represented by $\land $ and $\lor $ symbols, as you would know.

Here you can see its truth table.

$$\begin{array}{|c|c|c|} \hline p&q&p\longleftrightarrow q\\ \hline T&T&T\\ \hline T&F&F\\ \hline F&T&F\\ \hline F&F&T\\ \hline \end{array}$$

On the other hand, $\iff $ is used as a connective of propositional formulas. You can see both uses here: $$p\longleftrightarrow q \iff (p\to q) \land (q \to p)$$

And what does $a \iff b $ means? If you write $a\iff b $, then you could actually say the same by writing down that the bicondition $a \text { is true} \longleftrightarrow b \text{ is true} $ is always true. Note that this works whatever the truth values of $a \text { is true} $ or $b \text { is true}$ are.

Edit: in another fields a part of logic, (at least in basic degrees), choosing one or the other does not matter too much ($\longleftrightarrow $ or $\iff $ are just "lazy" math translations of simple English connector "if and only if").

As has been mentioned in the comments, this is almost certainly an idiosyncratic use, and the author (is that the right word for somebody who makes a YouTube video? Probably not) ought to have explained what he or she intended the symbol to mean. Without any additional context, it's hard to know for sure, but I'm going to hazard a guess that the symbol is intended to denote "are equivalent" in some (perhaps ill-defined) sense. In what sense? Probably in the sense of "equiconvergence" -- i.e., their convergence behavior is equivalent (one of them converges if and only if the other one does).