virtualbox.org

I downloaded Windows 8 yesterday from Microsoft's web site, at the offer price. The host computer is running Windows 7. Can I use Virtualbox to install W8 onto a VM and vice versa?

From a technical point of view, yes Virtualbox supports Windows 8.
From a legal and licensing point of view, nobody can give legal advice here, and I would suggest you read the Microsoft End User License.

Now this is said, from my understanding, you can upgrade machines (being physical or virtual) as long as there is already an licensed Windows on it.
Be aware that any cloning operation on the VM or use of the disk into another created VM will trigger the activation process again, and thus you will loose your license for that install.

Thankyou.

Is there a way to install it on a VM when I have downloaded it to my host desktop?

I will refer you to the user manual for a step by step on how to create a VM ---> here
Also, there is a thread about issues in general with Windows 8 as a guest ---> here