What is this symbol $\int$ called?

It is simply known as the "integral sign" or "integral symbol"; here is the Wikipedia article.

As others have already mentioned it is usually called the "integral sign" or the "integral symbol" or just "integral" for short.

Maybe more important is how the symbol shows up in context and how then to read it out loud.

For example $$\int f(x)\;dx $$ can be read as "the integral of $f$". Or: "the indefinite integral of $f$". With both of these it is fairly common to also say $f$ of $x$ instead of just $f$.

Note that even though the symbol might be called "integral" we usually will not use it in a sentence like "and then we take the $\int$ of the function and get.

See Integral Symbol. That is, it's usually called the "integral symbol".

For its origins:

"∫ symbol $\int$ is used to denote the integral in mathematics. The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz towards the end of the 17th century. The symbol was based on the ſ (long s) character, and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands. See long s for more details on the history of ſ."