One of the most frequently asked questions about Thunderbird's compose window is the contact pane. Mysteriously it disappears. Usually this is because you have pressed the F9 key and turned it off. The setting is to be found on the View menu as 'contact pane' and pressing F9 toggles it on and off.
The other cause is far less obvious, and that is it has been dragged to the far left of the window until the width is Zero. Recovery from this requires a bit of fiddling with the mouse over on the left centre of the screen to locate the now basically hidden divider and dragging it back to a visible location. The mail tweaks add-on contains an option to make these dividers flash when you move the mouse over them to make this process easier, but at the time of writing this the add-on had not been updated to support Version 6.