The accepted answer explains why you may not be seeing the described feature, but let me clarify the feature, itself.
Web sites offer varying support for auto-filling entries. They can use an @autocomplete attribute on input elements, and those inputs are identified by
field name. Otherwise, the input fields are just "dumb" places on a page to type stuff. In some cases, the web site does not want specific information auto-filled; password verification repeat entries is often an example.
Form filler applications can remember what content goes in what input locations based on things like input sequence and page location. Firefox has never had that level of form fill capability. It has always been limited to identified fields that support auto-completion.
What is changing is the mechanism. Since 2009, Firefox recommended auto fill entries based on frequency and recency. If it came across a named, auto-completable field, it would suggest for your selection, values you frequently use for an input field of that name. You would select from suggestions for each input field as you came to it.
What is in the process of being rolled out is the ability to use form profiles for all of the address, phone, and email input items. People often have alternate information they prefer to provide in different situations, and you can create multiple profiles that contain these common standard input items:
You can then select a form profile for a web site and input everything at once. It does require that you create the profiles. As you identified in the question, you select whether the feature is active and create profiles in Preferences | Privacy & Security. As David Postill's answer describes, this is in the process of being rolled out and is not yet available everywhere or in every version.
Mozilla is treating credit card information separately. This new feature looks like it may be available only in the Nightly version at this time (see Mozilla Support and Mizilla Wiki).
Further reading at: ghacks.net
