As an ex-CERN student, fellow and consultant at the time that the Web was being born I ordered "Weaving the Web" with great anticipation.
I was not disappointed. Tim explains in a very readable way the events that culminated in the creation of the Web and beyond.
His narrative style is excellent, especially as he describes his frustrations with CERN management (yes, I've been there!) and external organisations like NCSA, who attempted to usurp control over his creation.
My only criticism is that Tim is far too diplomatic in the book. I assume that this is because of his current obligations to the W3C - you can't really go around slandering W3C members :).
However, having said that, it is still possible to read between the lines and imagine the anger he must have experienced at times.
To summarise - a great read which most people interested in Web should enjoy.