Yes, it is because you used a copy of Vista from your computer. That is piracy, and illegal. You are allowed to use ONE copy of Windows on a SINGLE machine, and THAT IS IT. Anything else is piracy, and Microsoft has been adding teeth to this license provision that has been there since the birth of Windows.
You will need to go to some electronics store that sells copies of Windows and buy a copy of Vista to get that product key.
You could also call the Microsoft activation hotline, and they might be able to sell you a license, or at least transfer you to someone who can.
As for how much it will cost, that depends on the version you get, but figure on probably at least around $150US.
Finally, minor quibble, but you didn't download anything, you installed it. Downloading requires a network connection, and installing from local media on a local machine does not qualify. I mention it because it may confuse some and cause unnecessary delays in receiving assistance at some point in the future.
I downloaded Windows Vista on my mom's computer because she had several viruses on hers. It came with Windows XP in 2005 and it did not come with a disc to restore the system. It came with the paper that said to go through a certain process to restore it. BUT we couldn't go through that process since there were so many viruses on the computer.
The crux of the problem is that now the computer says that the windows is not genuine and needs a product key. Where do I find that and how much is it? Is it because I used the Vista from my computer? I'm sure that's probably it but any help/suggestions/recommendations would be appreciated.