I recently watched this education video about chemistry on YouTube about gas
At 8:52 there is an equation to determine how many moles of hydrogen gas there were in the Hindenburg: $$\frac{(100)(2.1189 \times 10^8)}{(8.3145)(283.15)} = 9.00 \times 10^6.$$ I don't understand how to calculate it. The only operations I know are multiplication (with the symbol $\times$), division (with the symbol $\div$), taking square roots ($\sqrt{}$) and exponentials ($A^B$), I know nothing beyond that.
First, could someone clarify what long line that separates the two rows of numbers does? I vaguely remember that it means the same thing as division but its confusing because there are multiple numbers both above and below it.
I also remember from school that we first need compute expressions inside parentheses, like $( 1 + 2 ),$ but in the equation in the video there are just single numbers in the parentheses and no mathematical operator sign like + - × ÷. Could somone explain this notation? I have no idea where to even start solving that equation.
What you are looking for is an explanation of basic math notation and order of operations. If you are interested in science videos, you may want to brush up on the basics and you can do so for free these days. I recommend the free prealgebra textbook at openstax.org, I have used it successfully to tutor students.
Now to answer your questions. The long line is just one of the ways to write division. If we have two numbers, $A$ and $B$, and $B$ is not zero (division by zero is not possible), then we have $$\frac{A}{B} = A \div B.$$ As an example, if $A = 44$ and $B = 11$ then we would have $$\frac{44}{11} = 4.$$
The parentheses are one way to write multiplication. If we have two numbers $A$ and $B$, all of the following are equal, they are just different (and very common) notations for multiplication: $$A \times B = A \cdot B = A(B) = (A)B = (A)(B) = AB.$$ The parentheses are simply used to make it more clear what is being multiplied. If I am working on a problem and I have been using vales $A$ and $B$, then it might not be confusing to write multiplication as $AB$. However, if I have two values $45$ and $21$, it would not be clear by $4521$ whether I mean the number $4,521$ or the multiplication $45 \times 21$. Therefore it is very common to write this multiplication as $(45)(21)$ or $45(21)$.
Including units could make it even more confusing, which is why they put each term into its own set of parentheses.
Finally, the operations of multiplication and division have the same order, so you can do them in any order you wish. However, it is probably easiest to do the multiplication in the numerator (the part on top of the division line) and the multiplication in the denominator (the part below the line), then you just have a simple two term division, like so: $$\frac{(100)(2.1189 \times 10^8)}{(8.3145)(283.15)} = \frac{211.89 \times 10^8}{2354.250675} = \frac{21189000000}{2354.250675} = 9000315.99226 \approx 9.00 \times 10^6$$
I have ignored any discussion of scientific notation (which is used here) or the idea of significant digits (used in scientific calculations) so you should probably familiarize yourself with those as well.