What do digital certificates really mean?

What do digital certificates really mean? The best discussion of this may be the one that I recently read in Peter Gutmann's book Engineering Security. Here's how Peter describes this:

As a pure speech act, what a certificate is saying is that at some point some entity who may or may not be the one named in the certificate probably requested that another entity who may or may not be the one named elsewhere in the certificate took the public components of a private key that the first entity may or may not control and asked the second entity to sign it using a private key that they may or may not control. However once it’s gone through many, many layers of software this has changed to (for example) a statement that the user has definitely connected to a web site controlled by the named entity, and by the time it gets to the user it’s jumped even further to become an assurance that it’s safe to enter sensitive personal and financial information on the web site!