What are some of the Websites/web-tools which must be a part of every mathematician's arsenal?

You probably want most answers to just give one suggestion, but I'll mention a pair that serve complementary aims. To convert LaTeX to an image, use codecogs; to find out the LaTeX command for something you draw by hand, use detexify (but you have to draw quickly).

Quiver lets you draw commutative diagrams easily and export to $\LaTeX$, and MathPix Snip lets you convert screenshots of equations to $\LaTeX$. Both very useful.

Also, libgen and scihub are indispensable with the modern publishing system...

The Math-aware search engine: Approach Zero is great for online investigations of mathematical expressions.

• Example: Looking for $[x^n]\frac{1}{\sqrt{1-x}}$ provides this result.