Either the present perfect or the past simple would work here. We might use the present perfect if the day's activities aren't over yet. For example, you might ask "What have you done today?" at lunchtime, where the implication would be "What have you done (so far) today?" You could also use the present perfect later in the day if you want to put the emphasis on your achievements. For example, at the end of the working day, you might say to your colleague "I've done A and B, so can you do C and D tomorrow?". [Tip: Note that the present perfect always tells us something about the situation now. So in the situation above, you're telling your colleague how much work has been achieved and how much still needs to be done] If the day's activites aren't going to continue, you'd be more likely to use the past simple, as 'What did you do today?" refers to a finished period of time. By saying 'What did you do today?", you're separating the activities of the day from the present moment. By asking the question in this way, you know that the other person's day of working, studying, sightseeing (or whatever) is over. [Another tip: If ever you're not sure which of these two tenses to use, you should always go for the past simple. In the majority of cases it will be right, or at least acceptable.]

The simple past and the present perfect are the two most difficult tenses to distinguish in English. Depending on the context, sometimes there is really no difference in meaning. Other times, there is a clear difference. Think of it this way, the present perfect reaches into the past and makes it still relevent in the present moment. In your example, there is no real difference in meaning. No difference in meaning: What did you do today? I went to the mall. (a finished action) What have you done today? I went to the mall. (a finished action) Difference in meaning: I lived in France for three years. (I don't live in France anymore.) I have lived in France for three years. (I still live in France.)

Generally we would use the present perfect if the day is still in progress (i.e. at lunchtime or in the afternoon) and the past simple if we are talking later at night. Depending on the context, either could be correct. In your answer, you can choose between the past simple and the present perfect. For example: A: What have you done today? B: I went to the doctor and went for a run. (once off actions) I've done most of my homework too. (action that is not completed) Remember, the present perfect is used to connect the past and the present. It can refer to unfinished time, an unfinished action, or an unfinished result.