'did' is the simple past
'done' the past participle.
- I did the washing up yesterday, now it's your turn.
- I have already done it!
The past simple is used to describe finished events, strongly bound to the past (by a date, a past event etc.). In the example above, 'yesterday' binds the whole sentence to the past, and calls for the past simple.
The present perfect lays the emphasis on the fact itself, not when it happened. In that sense, it's more bound to the present than to the past. In the example above, the second speaker don't say when s/he did the washing up; what's important is that it's done now.
As a rule of thumb, each time you have a reference to the past (yesterday, two days ago, when I was young etc.), use the past simple.