I've recently installed a VPN on my iPad (newest version) and when I start the VPN and try to use Google, I get a message that states I am unable to access Google due to my computer sending automated queries, which is against Googles Terms of Service. What does this mean, and how can I correct the issue?
Google generally does not track "computers" sending automated queries; it track automated queries by the IP address from which they originate. If you are using a VPN to connect to an external internet site, your traffic will appear to the site as if it originates from the outward connection point owned by your VPN. If another user of the VPN were to generate automated queries that were passed along by the VPN, their traffic would also appear to come from the same outward connection point, and thus have the same IP address as you.
I would suggest you take the issue up with your VPN provider; you are paying for the use of an IP address from them, but the behavior of other customers on the network has caused the reputation of that address to be tainted.
Report it to Google and they will investigate it: http://support.google.com/websearch/bin/request.py?&contact_type=ban