which file type to install my virtual hard disk in while installing a new machine on virtualbox?

VDI is the native format so it should be the ideal for virtualbox.

However, other forms allows compatibility with another system such vmware and "virtualpc"

• VDI = Native.
• VMDK = compatiblity with vmware.
• HDD = compatibility with Mac parallels.
• VHD = compatibility with "virtualpc" (and all latest microsoft products).

Windows 7 and windows 2008 r2 is able to mount VHD natively.

VMDK will be supported by VMware, and VHD is used by Microsoft VirtualPC. You can get more information here.

It really only matters if you're planning to use the same virtual hard disk with another virtualization provider.

If you're only planning to stay with VirtualBox then it really doesn't matter. But I would assume VDI would be preferred in this case as it seems to be the native format.

Even if you plan to move to another solution later on. There are ways to convert between the formats.