Difference between k-algebra isomorphism and k-extension isomorphism.

I am learning algebraic geometry. I know that $fcolon V rightarrow W$ is an isomorphism of algebraic varieties if and only if $f^{*} colon K[W] rightarrow K[V] $ is an isomorphism of $K-$algebras.

I have found the following theorem: Let $fcolon V dashrightarrow W$ a rational dominant map. $f$ is a birrational equivalence if and only if $f^*colon K(W) dashrightarrow K(V)$ is an isomorphism of $K-$extensions of $K$.

My question is, what is the difference between an isomorphism of $K-$algebras and an isomorphism of $K-$ extension of $K$?