As mentioned, the standard editing (like selecting and translation) can only be done with Objects. To add faces or to move individual vertices, an Object has to be in Edit Mode. 
You enter and leave Edit Mode by pressing TAB (or ALT-e or the header button). The cursor changes shape, selecting is now possible with individual vertices.  

Let's try this. Clear Blender (CTRL-x). Select the Plane (RightMouse click), and press TAB. Blender now is in Edit Mode.  
Selecting vertices is implemented identical as for Objects. 

- Select all four vertices of the plane: SHIFT-RightMouse, 
- press 'e': extrude? Yes. 
- Directly after extrusion, Blender enters 'grab' mode. Place the newly created vertices. Or:  
- If you are looking top-down: press ESC, go to front or side view (NumPad 1 or 3), press 'g' and place the vertices again. 
- Now you can repeatedly extrude-grab, extrude-grab, to create a snake-like structure. 

Extrusion is a simple and powerful method to create 3D objects. If the selection contains faces, these are duplicated. At the edges of the selection new faces are created. Blender extrusion intelligently deletes 'interior' faces. For example the mesh created in the previous example is completely hollow.  

Blender doesn't have an undo command. To compensate for this, Edit Mode always works on a copy. As long as you don't leave Edit Mode, the original data can be restored with key 'u'. 
 

In the Edit Buttons menu (use the header button or press F9) you can find a lot of tools that use extrusion, or subdivision, filling. These will be described later. 
 

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