When on the internet, does surfing with an account that has Administrator rights on the local machine you are on run more of a risk then going on the internet with a user who does not have admin rights on the machine?

Yes, it does.

This is my understanding of it: If you download and execute a piece of malicious software (which can be done without your knowledge), that software has the same rights as the logged-on user. So if you happen to be logged on as administrator it could format your hard drive etc. If you're logged on as a limited user then it can do much less damage - it might not even be able to run at all. Sometimes you have to log on as administrator and access the internet (e.g. to do critical updates) but don't visit any other web sites and log off again as soon as you have finished If you follow this advice, keep your antivirus up to date and use a firewall, then I think you will have no trouble.