Sensitive information should be hidden from prying eyes, both while being input and output, and when stored in the system. Sensitive information certainly includes credit card numbers, account balances, and home addresses, and in many applications also includes names, email addressees, and other private information.
Web-based applications should encrypt all communication with a user that includes sensitive information; the usual way is to use the "https:" protocol (HTTP on top of SSL or TLS). According to the HTTP 1.1 specification (IETF RFC 2616 section 15.1.3), authors of services which use the HTTP protocol should not use GET based forms for the submission of sensitive data, because this will cause this data to be encoded in the Request-URI. Many existing servers, proxies, and user agents will log the request URI in some place where it might be visible to third parties. Instead, use POST-based submissions, which are intended for this purpose.
Databases of such sensitive data should also be encrypted on any storage device (such as files on a disk). Such encryption doesn't protect against an attacker breaking the secure application, of course, since obviously the application has to have a way to access the encrypted data too. However, it does provide some defense against attackers who manage to get backup disks of the data but not of the keys used to decrypt them. It also provides some defense if an attacker doesn't manage to break into an application, but does manage to partially break into a related system just enough to view the stored data - again, they now have to break the encryption algorithm to get the data. There are many circumstances where data can be transferred unintentionally (e.g., core files), which this also prevents. It's worth noting, however, that this is not as strong a defense as you'd think, because often the server itself can be subverted or broken.