I think of logical necessity as (predictably enough) the necessity imposed by the laws of logic. So, for example, it's logically necessary that no proposition and its negation are both true, a necessity imposed by the law of noncontradiction. But one might regard logical necessity as broader than that, since one might say that it also includes conceptual necessities such as "Whatever is red is colored."

Metaphysical necessity is a bit harder to nail down. Every proposition that's logically or conceptually necessary is also metaphysically necessary, but there may be metaphysical necessities that are neither logically nor conceptually necessary, such as "Whatever is water is H2O" or "Whatever is (elemental) gold has atomic number 79." Nothing in logic or in the concepts involved makes those propositions necessary, but many philosophers say that those propositions are nevertheless "true in every possible world," which is the root idea of metaphysical necessity. Even if some proposition P isn't logically necessary, if P is metaphysically necessary then P is true in every possible world and the negation of P is false in every possible world.

This topic is actually more controversial than my answer suggests, but that's how I'd answer your question.