Blender Merge Vertices
The options to merge selections together at first and at last appear only when vertex select is the active mode of selection when in edit mode.
Blender merge vertices. Every island of selected vertices connected by selected edges will merge on its own median center leaving one vertex per island. In the merge pop up menu that appears select at first at last at center at cursor or collapse. Shift select the cube and plane hit ctrl j to join the two objects. So now when you select either object both objects will be selected.
To merge elements together first make a vertex edge or face based selection then from the vertex menu click merge vertices alt m vertex merge vertices. The button in in the left tool shelf hotkey t and when you press it you get a panel for setting the merge distance and up in the info panel header you can see the removed nnn vertices information. I think arexma explained it quite well but you seemed to have missed the point that the two objects must be one object. Every island of selected vertices connected by selected edges will merge on its own median center leaving one vertex per island.
If this is the case then you can enter edit mode and join vertices. Merging vertices of course also deletes some edges and faces. Before this update the only thing you needed to do was to press alt m but that brings up another menu now. It is also available via the mesh edges collapse menu option merging vertices of course also deletes some edges and faces.
But blender will do everything it can to preserve edges and faces only partly involved in the reunion.