The prompt for your personal password/number is nothing to be alarmed by, quite the opposite actually - it's something that should make you feel safer. It is part of a new security standard that's being implemented across the world, mainly by VISA and MasterCard (who, of course, have their respective names for this; 3-D Secure vs. SecureCode). To ensure that nobody but the rightful owner of a credit card tries to use it for purchases, the personal password/number must be entered before the transaction can be completed (it's actually up to the bank to decide how you identify yourself but the personal password/number seems to be the most common option).
As this security standard is being implemented, cardholders will be issued new credit cards along with personal passwords/numbers for them and they should also be informed by their bank about the added security this brings. When the issuing banks have their part of 3-D Secure all set up, these consumers will then be asked to authenticate themselves the way the issuing bank has decided.
The PIN prompt will of course only be shown to a person that has a card that is new enough to be part of the new security standard. If you still don't feel at ease about all this, contact your card-issuing bank and they'll be able to tell you more about it.