You are using the Gnome-Classic interface - either you are using this by choice, or you are using the fallback mode which occurs if your graphics card & driver doesnt the 3D Acceleration required for the full Gnome-Shell GUI.
To add and remove application launchers in the gnome-panel you need to:
Press Win+Alt and right-click the top menu bar - N.B. Win is the Windows Symbol key
If you are not using Compiz then you need to :
Press Alt and right-click the top menu bar
Much more information is described in the linked Q&A and other links in that answer.