The Layer Manager is used to manage the order of display layers within a view - the top most layer in the list is drawn over all other layers.
Each layer is listed with it's name (typically the file it comes from), an eye icon ( - if visible) and a trashcan symbol (). Clicking on the eye will cause the layer to toggle between being visible, and not being visible. Clicking the cross will remove the layer from the view.
Left clicking on a layer will cause it to become the active (highlighted) layer. Many operations such as enhancements, and vector editing operate on the selected layer of the selected view.
The icon can be used to create a new vector layer for digitizing into.
The icon will cause currently selected layer to move up in the layer dialog, causing it to be drawn over the layer that was previously above it. The icon will cause currently selected layer to move down in the layer dialog, causing it to be drawn under the layer that was previously below it.