Why are the vertices extruded instead of the edge? I selected the two ending vertices of an edge to extrude it but instead of getting the entire edge extruded (after selection) I got only the vertices extruded? I've met this problem a couple of times before. How do I solve it?

Here's a screenshot of the extruded vertices:
You need to select the edge and then extrude. Select the edge using the icon marked in yellow.

EDIT:
When the vertices option is selected you can perform actions on the vertices them self. So you can extrude every vertice separately, or like in your case the ones you selected. When the edge option is selected you can perform actions on the edges. And the same for faces.