The Girl Who Played With Fire until just now.
Like the first one, it had a wonderfully complicated plot and a more than average dose of who-done-it. It had the same main two characters, Mikael and Lisbeth, as well as some of the same supporting characters. I really liked how much more you got to know Lisbeth in this book. She is so much easier to relate to and given the plot, it's important for the reader to "get" her. I won't give anything away but just know, her history is worth paying very close mind to.
One part I found odd was the very beginning. Lisbeth is in the Caribbean and a hurricane strikes. She develops a few relationships, a couple positive and one negative. However, once she leaves the Caribbean, those people don't tie back in. The only thing that makes sense about that part to me is helping illustrate Lisbeth's characteristics, morality, etc. Kinda odd though.
It has some obscure (well obscure to me) references to Swedish history but the footnotes are concise and clear. Thus the historical references didn't trip me up and I can see how a Swedish audience would connect to them.
As the second in the series, it makes sense to read it second. I think you could get away with having not read the first but there is quite a bit of building. To enjoy the book the most, I suggest reading The Girl With the Dragon Tattoo first. And on the same note, for optimal enjoyment, keep in mind, this book centers around a murder. I found I couldn't really read it on nights when I was home alone. (But when my big strong hubby was home to protect me from things that go bump in the night, no problem. Bring on the murder mysteries!) If things like that make you a little jumpy too, I'd consider it when picking this book up.
Finally, also like the first book I have one complaint: the similarities in characters names. I know this sounds trivial but there are a lot of characters and I swear half of their last names start with B. I had to really think sometimes about who was the cop, who was the crooked lawyer, etc. Some variation in names and spelling of names would have greatly helped in keeping the myriad of characters straight.
Verdict: read it but begin at the beginning by reading the first one first. And maybe consider taking notes on the many characters named B.