When I delete a picture the picture disapears but a black image stays in the place where the picture used to be.
I think your problem is simple: You apparently deleted the picture while outside Lightroom. Then there should be a little exclamation point in the upper right corner of the thumbnail's border when you're in grid mode. All you have to do to get rid of this problem is to select the black thumbnail and delete it (Lightroom might ask you if you want to discard it). To prevent this is the future, just do your deletions while inside Lightroom - then it updates its catalog by removing references to that item while removing the item from your sytem.