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There are two main ways to have subtitles in an AVI movie.
The first one is to include the subtitles in the file itself, so they will be always visible. The other way is to save subtitles in an external file (there are many different formats, one of the most diffused is the .srt that is basically a text file containing all subtitles and all the needed information to display them correctly).
If subtitles are embedded in the avi, they will be shown always, and you can't disable them. With an external file you can choose if show subtitles or not.
To include subtitles directly in the file the easiest way is to encode directly the AVI with subtitles from the original DVD. AutoGK allow this very easily.
To extract subtitles from the original DVD and then create the .srt file you can use the free (and excellent) tool DVDSubEdit.