About Pivots

How can I modify a pivot of an object? What are all these pivot-types (Median Point, Invidual Centers etc.)? Do they have some other functionality, than that the object scales and rotates in relation with them?

You can modify the pivot by selecting the object then pressing F9.

You can Center, Center New and Center Cursor.
Center will center the object on the current pivot point.
Center New will move the pivot point to the center of the object.
Center Cursor will move the pivot point to the position of the 3D Cursor.

Try this…
Restart Blender so you have the default scene with the cube.
Select the cube.
Press F9.
LEFT-Click in the 3D view. This will move the 3D cursor from the origin of 0,0,0 to wherever you click.

Click Center cursor and the pivot point will jump to the 3D cursors position.
Click Center and the cube will jump to the position of the pivot.
LEFT-Click again to pick a new position for the 3D cursor.
Click Center Cursor.
Then click Center New and the cursor will jump to the center of the cube.

When I press F9 nothing happens. I got version 2.45.

When I press F9 nothing happens. I got version 2.45.

This hotkey puts the buttons window in the editing context which is shown by default when blender is opened. You are probably already in the right place.

The OP was asking about pivoting options. The pink dot you’re all discussing is the “Object Center” and it’s not necessarily always the pivot point. You can make an object pivot about a virtual point that’s separate from the object center, or even revolve about a point without itself rotating. The best explanation I’ve seen for controlling pivoting is this video tutorial at Blendernewbies. They do an especially good job with explaining “move individual object centers.”