Verified by Visa is an implementation of 3D-Secure, a technology built by Visa that adds an additional layer of security for your credit card transactions.
It's basically a secondary password registered with the Visa servers that has to be entered along with your Date of Birth for verifying the authenticity of your transactions.
So in worst case scenarios, if your card number is acquired by anyone with malicious intentions, he/she cannot use it for any internet transactions without going through the verified by visa. The catch, however, is that if the person knows your ATM pin, the VBV password can be reset on the fly within seconds.
Mastercard has adopted the same technology and calls it Mastercard SecureCode, JCB Cards also utilize the 3D Secure technology and call it J-Secure and Amex's implementation of the same is called American Express Safekey.
You can read about it at the following links -
3-D Secure - Wikipedia, the free encyclopedia
https://cardsecurity.enstage.com/ACSWeb/EnrollWeb/SBI/main/vbv.jsp
MasterCard SecureCode | Credit Card Security Code | MasterCard®