If you are new to programming or if you have experience in other languages and are new to C++, these books are highly recommended.
1.C++ Primer† (Stanley Lippman, Josée Lajoie, and Barbara E. Moo) (updated for C++11) Coming at 1k pages, this is a very thorough introduction into C++ that covers just about everything in the language in a very accessible format and in great detail. The fifth edition (released August 16, 2012) covers C++11. [Review]
2.Accelerated C++ (Andrew Koenig and Barbara Moo) This basically covers the same ground as the C++ Primer, but does so on a fourth of its space. This is largely because it does not attempt to be an introduction to programming, but an introduction to C++ for people who've previously programmed in some other language. It has a steeper learning curve, but, for those who can cope with this, it is a very compact introduction into the language. (Historically, it broke new ground by being the first beginner's book using a modern approach at teaching the language.) [Review]
3.Thinking in C++ (Bruce Eckel) Two volumes; second is more about standard library, but still very good
4.Programming: Principles and Practice Using C++ (Bjarne Stroustrup) An introduction to programming using C++ by the creator of the language. A good read, that assumes no previous programming experience, but is not only for beginners.