Topic Inherit - If selected this means that the article will be inherited by the parent topic. If not selected then the article will be viewed only in the selected topics.
Topic Default - If the article has a number of topics it belongs to then the default topic is the one used when Geeklog doesn't know what topic the visitor is in. The Topic the visitor is in is stored as part of the session. This could happen for example if someone visits that article page first. Since the topic has not been set yet it will use the article default.