Duplicate Layer: This option will copy the layer you have selected and make an exact replica of it as a new layer.
Delete Layer: This will delete the layer you have selected in your layers palette. Unless you turn the warning off, Photoshop Elements will ask you if you are sure you want to delete the selected layer.
Rename Layer: This allows you to select a layer and rename it something more fitting rather than the default layer 1, layer 2, layer 3, etc…