Blender can be used to edit a mesh by deleting vertices, edges or faces from it.
Edit Mode. Switching the Mode can be done from the dropdown in the bottom bar.
Vertex select modebutton.
Shiftkey pressed, if you want to select many elements. You can only select the elements of the select mode you picked. For example, in Vertex select mode you can only select vertices.
Deletekey. A context menu comes up and you can choose what you want to delete: vertices, edges, faces or all. For example, if you chose a vertex, the dialog is asking whether you want to delete that vertex, or the edges incident to that vertex or the faces incident to that vertex or all of the above. Select what you want and those elements will be deleted from the mesh.
Tried with: Blender 2.62 and Ubuntu 12.04 LTS