My guess is that that character is a typographic error and the author should have written \verb|'\"'|, producing:

since this is indeed a way to specify one character constant containing the character double quote ("), in C language (in fact, the \ is superflous in this case, but nevertheless it can be put too). I could verify my guess if you provide more context about the part in the book in which that fragment appears.
My main concern now is, how in the world the author of the book managed to type a diaeresis (¨) when he intended to type a double quote (")?
One possible answer is that he doesn't know the difference, which seems unlucky in a book about programming, and even in this case typing a ¨ would be more difficult than typing a ".
Another possible answer is that he writes in an editor configured to use the quote and double quote symbols as diacritics. That is, if he writes ' followed by e, he gets é, or if he types " followed by o he gets ö. In this case to get " he probably would have typed " twice, but instead typed " followed by a space (or by ') and got ¨.
Finally, a third option, and the most probable IMHO, is that instead of using verbatim to typeset that fragment, the author used the following command:
\texttt{'\textbackslash"'}
This produces:

Which still does not show the diaeresis, but it shows a backslash more similar to the one in your figure.
How the diaeresis could come? I think that the author was using some babel package for foreign languages. It is common that those packages make " an active char in order to facilitate the input of diacritics. For example, spanish package makes so, and when you type "u in the source you get ü in the document. However, the combination "' issues an error with spanish, so that could not be the language the author used. Perhaps you can clarify in which language the document is written.
Addendum (off topic)
About the use of the unicode character U+00A8 in the modern world, I have to say that, despite the fact that letters such as ö, ü, etc. have their own Unicode point (which is U+00F6 and U+00FC respectively) the Unicode standard also allows their codification as the sequence of two unicode chars: o¨ and u¨, respectively (U+006F, U+00A8 and U+0075, U+00A8). So U+00A8 is not entirely useless. In fact, OSX uses this way of coding diacritics in the name of the files and folders in its filesystem! (which causes some troubles).
\"{}is what you're looking for? It's a diaeresis, which is usually placed above vowels to make them umlauts (as inüorÄ), in which case you'd say\"uor\"A. – Jake Dec 12 '12 at 20:47\"{}gives you two dots. – Jake Dec 12 '12 at 20:50