In a nutshell, TrackBack was designed to provide a method of notification between websites: it is a method of person A saying to person B, "This is something you may be interested in." To do that, person A sends a TrackBack ping to person B.
A better explanation is this:
The idea here is that more people are introduced to the conversation (both Person A's and Person B's readers can follow links to the other's post), and that there is a level of authenticity to the trackback comments because they originated from another weblog. Unfortunately, there is no actual verification performed on the incoming trackback, and indeed they can even be faked.
Most trackbacks send to Person A only a small portion (called an "excerpt") of what Person B had to say. This is meant to act as a "teaser", letting Person A (and his readers) see some of what Person B had to say, and encouraging them all to click over to Person B's site to read the rest (and possibly comment).
Person B's trackback to Person A's blog generally gets posted along with all the comments. This means that Person A can edit the contents of the trackback on his own server, which means that the whole idea of "authenticity" isn't really solved. (Note: Person A can only edit the contents of the trackback on his own site. He cannot edit the post on Person B's site that sent the trackback.)
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