This is from A field guide to Algebra by Antoine Chambert Loir.
A polynomial $P \in K[X]$ is separable if and only if its roots are in an algebraic closure of $K$ are simple.
Here is another definition.
An irreducible polynomial $f \in F [X]$ is separable if $f$ has no repeated roots in a splitting field. It is called inseparable otherwise. Note that if $f$ is not necessarily irreducible, then we call $f$ separable if each of its irreducible factors is separable.
Why two different definitions?