It's a good idea to get an understanding of objects in Blender. Your example shows an object in blender and the origin of that object is the large orange dot. When you move the object, the origin will move with it.
Now your object contains mesh information too. When you move your object, the mesh information moves too. However you CAN edit the mesh information by hitting the tab key to go into edit mode. While in edit mode, you can manipulate the mesh data belonging to the object, BUT NOT THE ORIGIN. This is what you are seeing in your example; you are in edit mode moving the mesh, but not the object.
If you wish to move the object, get out of edit mode by clicking the tab key again, then move your object.
It's also unclear by your question if you understand the difference between the cursor and an object origin. For information on the cursor see Blender 3D Cursor