Apparently, Office 2007 no longer has character overwriting enabled by default. Pressing the Insert key on your keyboard does nothing at all. While most rarely use this function, a customer was blaming his keyboard from nothing working properly. A quick look around showed that, indeed, pressing Insert no longer enables the overwrite function in Office 2007 (Word in particular was tested).
Microsoft has decided some time ago to turn off this function by default, because it is indeed rarely used - it's the first time I noticed it was gone, and have never received a question up until now, and because there were complaints about users inadvertently turning on overwrite mode and overwriting their precious text.
Luckily for those that still use this function, Microsoft has not entirely disbanned the feature, but has rather stowed it away in the Options window of the individual applications.
For Word, all you need to do is navigate to Word Options, then head to Advanced and tick the box as shown below.
That's all there is to it.
If you want overwrite mode to be enabled by default, you can also tick the box Use overtype mode.