I have BOX MS Office 2010 and run it on my Windows machine. Can I install it/migrate licence to OS X? I do own BOX version of MS Office 2010.

After switching to OS X is it possible to migrate my licence to the OS X version of Office or if not needed than to install it? 

Is it multi-platform or per-platform product?

I've red some Microsoft paper about it - an official documentation - but it's more enterprise oriented with volume purchuases.
No, you cannot migrate your license from Office for Windows 2010 to Office:Mac 2011.  You will need to purchase a new license for Office:Mac.  

You could run Office for Windows on your Mac.  If you have a license for Windows, you could install Windows either using Boot Camp or a virtualization application like VMware Fusion or Parallels.  After you have installed Windows on your Mac, you can then install your existing copy of Office for Windows.