Learn Danish: Jeg kender manden, men jeg ved ikke hvad han hedder. - I know the man, but I do not know what he is called.

This shows the exact difference between the two verbs! Use kende (here as kender) when you are familiar or acquainted with a person, place, or thing. Use vide (here as ved) when you know a fact or a piece of information. You are acquainted with the man (kender), but you do not know the fact of his name (ved).

Why is it ved ikke instead of ikke ved?

In a main Danish clause, the adverb ikke (not) usually comes right after the first verb. Even though it follows the word "but" (men), jeg ved ikke acts as a main clause, so ikke goes directly after ved.

Literally, doesn't hvad han hedder mean "what he is called"?

Yes! In English, we usually say "I don't know what his name is". In Danish, it is much more common to use the verb hedde (to be called/named) rather than using the noun for "name" (navn). So it translates perfectly to "what he is called".

If I just want to ask "What is his name?" on its own, do I say hvad han hedder?

No. When you ask a direct question, the verb must come in the second position, right after the question word: Hvad hedder han? (What is he called?). However, in our example sentence, it is an indirect question built into a larger sentence, so the subject sneaks in before the verb: hvad han hedder.