Your first sentence has got a missing word. You can say "What have you done today?" or "What did you do today?" and both are perfectly fine.

"What have you done today" is of a degree heavier in tone though--suggesting blame, as in "List the tasks that you've accomplished today."

Both are correct. however, in different contexts. If you consider, that the day is almost over and nothing more should be done on that particular day you will ask "What did you do?". If its only midday and the person still may do some more things to say, ask "What have you done today?"

"What have you done today?" (correct) "I have done a lot of work today." We use the present perfect tense for a period of time that continues until now (unfinished) "What did you do yesterday/last week?" "l did a lot of work yesterday." (a finished time in the past)

What did you do?