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Significance
A DNS error means that the PSP will not be able to connect to the Internet. It most commonly indicates a problem connecting to the wireless router, not a problem with the router's connection to the Internet itself. The problem will then prevent the ability to play online games, to update the PSP's firmware (or operating system), to surf the Internet with its built-in browser, or to visit the PlayStation Store.
Types
The PSP's DNS error will most often be associated with a particular error code (80410410) which means that, although a PSP has found a wireless router, it cannot make a stable connection. Errors of this type are easily fixed. If, however, the PSP experiences a DNS error while also being unable to locate an otherwise functioning router, then there is most likely a problem with the PSP's hardware.
Solution
Fixing the problem involves checking both the settings on the wireless router and on the PSP. For the router, check the user's manual to make sure the following settings are correct: SSID (service set identifier) should be turned on, the WEP (wired equivalent privacy) should be reset, and you should make sure that the WEP key/password is available to the PSP. (Turning the WEP off, or making your wireless router available to any connection, is also a last-ditch option.) On the PSP, make sure that the WLAN switch on top of the PSP is on (to the right), and that the IP address the PSP tries to use to connect to the router is correct. You can manually input 0.0.0.0. for this if you do not know how to locate your router's IP address.
Misconceptions
Many people think that a DNS error means that the PSP has suffered a catastrophic error. This is not the case, and the error only means that the PSP is having a temporary problem which can quickly be solved by taking a look at the hardware settings involved.
Tip
Sometimes a DNS error can occur when you simply have a poor signal connecting the router to the PSP. Disconnecting the PSP from the Internet or turning off the WLAN switch and then reconnecting it can sometimes increase signal strength.