Does the fiber product of two normal varieties remain normal?

Are you asking about products or fiber products? If you're asking about fiber products, the answer is no. For example, you can have two smooth surfaces in A³ whose intersection is a nodal cubic (see the picture on this page of Hartshorne). This intersection is the fiber product of the two surfaces over A³.

The answer is yes.

In general one can define a normal morphism of schemes $f:X \rightarrow Y$ to be a flat morphism such that for every $y \in Y$ the fibre over $y$ is geometrically normal.

Then we have the following theorem on normality and base change (see EGA Ch 2 IV 6.14.1)

Let $g: Y' \rightarrow Y$ be a normal morphism of locally noetherian schemes. Then for every normal $Y$-scheme $X$ the fibre product $X \times_Y Y'$ is normal.

Over an algebraically closed field flatness and geometric normality reduce to just being normal so the result follows.