C++ FAQ Celebrating Twenty-One Years of the C++ FAQ!!!
(Click here for a personal note from Marshall Cline.)
Section 27:
[27.4] What's the difference between <xxx> and <xxx.h> headers?

The headers in ISO Standard C++ don't have a .h suffix. This is something the standards committee changed from former practice. The details are different between headers that existed in C and those that are specific to C++.

The C++ standard library is guaranteed to have 18 standard headers from the C language. These headers come in two standard flavors, <cxxx> and <xxx.h> (where xxx is the basename of the header, such as stdio, stdlib, etc). These two flavors are identical except the <cxxx> versions provide their declarations in the std namespace only, and the <xxx.h> versions make them available both in std namespace and in the global namespace. The committee did it this way so that existing C code could continue to be compiled in C++. However the <xxx.h> versions are deprecated, meaning they are standard now but might not be part of the standard in future revisions. (See clause D.5 of the ISO C++ standard.)

The C++ standard library is also guaranteed to have 32 additional standard headers that have no direct counterparts in C, such as <iostream>, <string>, and <new>. You may see things like #include <iostream.h> and so on in old code, and some compiler vendors offer .h versions for that reason. But be careful: the .h versions, if available, may differ from the standard versions. And if you compile some units of a program with, for example, <iostream> and others with <iostream.h>, the program may not work.

For new projects, use only the <xxx> headers, not the <xxx.h> headers.

When modifying or extending existing code that uses the old header names, you should probably follow the practice in that code unless there's some important reason to switch to the standard headers (such as a facility available in standard <iostream> that was not available in the vendor's <iostream.h>). If you need to standardize existing code, make sure to change all C++ headers in all program units including external libraries that get linked in to the final executable.

All of this affects the standard headers only. You're free to name your own headers anything you like; see [27.9].