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Essential C++ by Stanley Lippman Recommended
This is a really good book for getting started with C++. Its a thin little book that tells you what you need to know, without going crazy. Its more about the language and how to program in it than it is about a philosophy of how to organize your program (which many books seem to do).

(Buy at Amazon) (Publisher Site)

The C++ Primer by Stanley Lippman, Josee Lajoie and Barbara Moo Recommended with Caveats
(Buy at Amazon) (Publisher Site)

When I learned C++ (1989), I went to the bookstore and there were 2 books. So when I tell you the book that I used to learn C++, you have to take the recommendation with a grain of salt.

The book I learned from (and is probably my favorite C++ book) is C++ Primer. (published by Addison Wesley). With each version, it seems to double in size (this time it added a second author). When I got the first version it was a compact little volume. To be honest, I think it gets worse as it grows bigger.

With the 4th edition, the "getting bigger" has gotten to a crisis - there's so much in here, and fancy stuff is mixed in with the basics. Part of this is that C++ (the language) has gotten so big and complicated that there really is 900 pages of stuff to talk about.

Even the author laments the fact that a comprehensive book on C++ is unwieldy, which is why he wrote Essential C++