It seems like some of the app images on the Windows 8 start screen have a "Larger" button to show a rectangle bigger layout tile (compared to the square). But I noticed that a number of apps don't have this option when I right click on the icon. I can't see any pattern here. Is there some logic why some app can be shown in the larger tile and some can only be shown in the square size ?
So basically, it's up to the app developer to implement this optional large tile mode.
Only "metro" apps, apps that support the metro interface have that options. So most apps you download from the Store will have that.
On the other hand, traditional desktop application that are not gear towards the mobile interface, do not support the metro interface, will only show a enlarged icon.
Update after discussion with ioSamurai
Additionally, just as ioSamurai pointed out in his answer, "metro" app may or may not support the large tile style. That is completely a developer choice.