The patient wakes up after surgery and ask "what is my name?" :))

як = how
мене = me (same reaction to я as between I and me in English, in this case, in that it's "[someone] calls me")
звати = to call, and you can also use звуть, they call

So literally, "how do they call me/how to call me" = "what is my name?"

Just in case knowing how the nuts and bolts works helps you :)

Як мене звати?

Now tell me what my name is.

You know, when you're knocked out and wake up in Ukraine and you've lost the information that tells you who you are, but not your Ukrainian vocabulary. That's when.

Звати is the infinitive (not conjugated ) version. Other one is conjugated for "they", see above comments.

There is no difference, звать is more informal, звати is formal

I ended up saying something like this in Russian on lang8, asking which translation of "Andrew" I ought to use!