What the Error Means
A DNS error 807101102 means that the PlayStation 3 has stopped being able to communicate with your Internet connection and is attributing it to an error in your DNS. DNS stands for "Doman Name System" and is essentially the tool your computer uses to communicate with your Internet service provider to allow online content to appear on your various devices.
What Causes the Error
It isn't clear what specifically causes this error. It is likely a quirk with the PlayStation 3 firmware and its interaction with your specific router or Internet connection. There is no real way to predict its occurrence or to trigger it intentionally.
How to Fix It
You will need to manually enter the DNS domains, default gateway and subnet mask from your PC into the network settings on your PS3. You will also need to assign a static IP to the PS3 in these manual settings. IP addresses assigned to your various devices tend to change as routers are reset and devices are taken off and on the network. Assigning a static IP will ensure a stable connection to your router and avoid future DNS errors.