This is the John Keegan one. I was surprised when I started reading this and it said it wasn't a complete history. It looked pretty big to be a non-complete history. It was one of the books my history teacher recommended, and I thought I might as well get hold of a copy. I didn't bother with the A-level books because I'd got the ones from that series on Russia and I thought they were pretty rubbish--they had definitions for all the words I understood, and no definition for the odd word they used that I didn't really know what it meant. Which was annoying as you can imagine. Anyway, I read this book, and I found it really interesting.
Unlike the Osprey books, it did a good job of focussing on the military stuff like I expected it too. It was very well structured, with the war split into six sections, based on time and theatre. I found the chapter on the 'Resistance and Espionage' very interesting--basically Keegan said that the resistance movement was pretty useless and just wasted lives. That did surprise me--I'm sure they served some purpose (although thinking about it, I don't know what precisely... Maybe they didn't.). However, he does say that the 'sigint' (signal intelligence, basically Ultra) section was a lot more useful.
The sections which focussed in on battles were also very interesting. I enjoyed reading about the carrier battle (I never really realised that Midway was the first...). And there was the Battle of Britain, tank battles, city battles, all sorts. I can highly recommend this book. There was also a lot about the strategy and the decisions that had to be taken regarding it (for example relating to the opening of the second front, and how Churchill was desperate to push it back to 1944 in order to get the build up and preparations sorted so as to minimise casualties). I thought it very balanced with regard to which bits he chose to focus in on, and with regard to the coverage of both sides.
Anyway, I would highly recommend it as a very good general history of the Second World War. Maybe it isn't entirely comprehensive, but all the major bits are covered in good detail, and it's well written. I read it through like a proper book, but it could also be used for reference (I have a feeling I'll be rereading bits and pieces when we start studying the actual war rather than just the build up in history).