You want to know which algebras $A$ are such that $A\otimes B$ is central simple. (All algebras and tensor products being over $F$.) If $Z(A)$ and $Z(B)$ are the centres of $A$ and $B$ then $Z(A)\otimes Z(B)$ is contained in the centre of $A\otimes B$. Hence $A$ and $B$ must both have centre $A$. If $I$ is a two-sided ideal of $A$ then $I\otimes B$ is a two-sided ideal of $A\otimes B$. As $A\otimes B$ is simple, then $I$ is zero or $A$, that is $A$ is simple.

We conclude: the only algebras in $X$ are the central simple algebras over $F$.

You want to know which algebras $A$ are such that $A\otimes B$ is central simple for some algebra $B$. (All algebras and tensor products being over $F$.) If $Z(A)$ and $Z(B)$ are the centres of $A$ and $B$ then $Z(A)\otimes Z(B)$ is contained in the centre of $A\otimes B$. Hence $A$ and $B$ must both have centre $F$. If $I$ is a two-sided ideal of $A$ then $I\otimes B$ is a two-sided ideal of $A\otimes B$. As $A\otimes B$ is simple, then $I$ is zero or $A$, that is $A$ is simple.

We conclude: the only algebras in $X$ are the central simple algebras over $F$.