Can I update my operating system and still keep my Mac mini

What do you mean "and still keep my Mac mini"? If you didn't "keep" it, what, exactly, would you update the operating system on?

If you are still running 10.6.6 you should first update to :

Mac OS X 10.6.8 Update Combo v1.1

Then you could upgrade to Yosemite but you may lose some of your older software compatibility.

If you have enough room on your hard drive you could create a partition and install Yosemite on the new partition. Then you could select which OS you want to use. You should have at least 4GB of RAM installed 8GB would be better. That way you can try out the new OS and see if it meets your needs. OS X 10.6.8 is a very stable system but its life cycle is coming to an end as it is not officially supported any longer so there are no new updates specifically for it.

That that the early and late 2006 Minis can only go to Snow Leopard (10.6)

The 2007 can only go to Lion (10.7)

How to identify Mac mini models - Apple Support

Later ones can go to Yosemite

http://www.apple.com/osx/how-to-upgrade/

Also not that if yo to later than Snow Leopard some apps may not work. See:

Upgrading to 10.7 and above, don't forget Rosetta!