They are defined differently.
Suppose we have a field $K$.
We say a finite field extension $L$ is separable over $K$ iff the number of embeddings $L\hookrightarrow \bar K$ into the algebraic closure $\bar K$ which is invariant on $K$ is $[L:K]$.
We say a finitely generated $K$-algebra $A$ is separable iff for any field extension $E/K$, $A\otimes_KE$ is reduced.
I have read that separable algebra is a generalisation of separable field extension. I think there is a proposition that
For a finite field extension $L/K$, if the field extension is
separable, then for any other field extension $E/K$, the $E$-algebra
$L\otimes_KE$ is reduced.
Can someone give some suggestions, I have no idea how to apply separability of field extensions. Or even a simpler case about $K[\alpha]$ for $\alpha$ separable over $K$.