"Wait" is an intransitive verb, so it doesn't need an object. "I wait," is a complete sentence. So, to add extra information you need prepositions. "For X" indicates the reason why one waits. "Wait for me" is the only correct sentence. Keep an eye on those transitive and intransitive verbs and things should go fine.

( wHich one is correct? ;) 'Wait for me' is correct.

Which one is correct/right? The second one."wait for me"

No, wait me is not making proper sense."wait for me" is right. So 2nd one is right.