Cloud = "through the Internet"
Personal, in this context = "from your house using your home Internet connection"
It's not necessarily grid computing. Grid computing is making multiple machines act as one, allowing their combined resources to be utilized in parallel. Some Internet accessible services, such as web servers, do use concepts similar to grid computing, such as clustering, load balancing, etc, in order to scale up to be able to service millions of users.
Probably what your friend meant, is that if you set up a RAID and NAS at home, there are ways to make the data on that RAID/NAS accessible via the Internet, through your home Internet connection, from any other Internet connected computer or device. A very simple (and bad) example is setting up a publicly accessible FTP server at your home, pointed to the root directory of your NAS.
Storage doesn't involve much computing unless you are talking about encryption or erasure encoding (the part of RAID that computes parity).