I have received an old MacBook from my Dad and he had wiped it clean, but I don't think he's done it correctly.
I am having issues with unmounting and re-installing Lion OS X back on, which it came with.
Can anyone help me please?
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Probably something went wrong wiping the disk. You can't properly erase/repartition a disk the OS X booted from. Additionally it seems that you don't have a proper Recovery HD.
I recommend either to create a bootable installer thumb drive with the Lion installer or booting to Internet Recovery Mode by pressing altcmdR while booting, formatting the internal disk and restoring OS X. Not all Macs boot to IRM though.
To create a bootable thumb drive you need the Lion installer app, a second Mac and a USB thumb drive (8 GB or more)
You can't unmount the disk because you're booting from it. You'll need to boot from an external volume or the internet: I would recommend the latter since it's probably the fastest and most hassle free for trying to reinstall whatever it came with. However, not all Macs support this (yours probably does), so check and see if it's compatible. If not, install OS X on an external volume and boot from it.