.c vs .cc vs. .cpp vs .hpp vs .h vs .cxx

It really doesn't matter.
If you feed .c to a c++ compiler it will compile as cpp, .cc/.cxx is just an alternative to .cpp used by some compilers.

.hpp is an attempt to distinguish header files where there are significant c and c++ differences. A common usage is for the .hpp to have the necessary cpp wrappers or namespace and then include the .h in order to expose a c library to both c and c++.

Those extensions aren't really new, they are old. :-)

When C++ was new, some people wanted to have a .c++ extension for the source files, but that didn't work on most file systems. So they tried something close to that, like .cxx, or .cpp instead.

Others thought about the language name, and "incrementing" .c to get .cc or even .C in some cases. Didn't catch on that much.

Some believed that if the source is .cpp, the headers ought to be .hpp to match. Moderately successful.

Historically, the first extensions used for C++ were .c and .h, exactly like for C. This caused practical problems, especially the .c which didn't allow build systems to easily differentiate C++ and C files.

Unix, on which C++ has been developed, has case sensitive file systems. So some used .C for C++ files. Other used .c++, .cc and .cxx. .C and .c++ have the problem that they aren't available on other file systems and their use quickly dropped. DOS and Windows C++ compilers tended to use .cpp, and some of them make the choice difficult, if not impossible, to configure. Portability consideration made that choice the most common, even outside MS-Windows.

Headers have used the corresponding .H, .h++, .hh, .hxx and .hpp. But unlike the main files, .h remains to this day a popular choice for C++ even with the disadvantage that it doesn't allow to know if the header can be included in C context or not. Standard headers now have no extension at all.

Additionally, some are using .ii, .ixx, .ipp, .inl for headers providing inline definitions and .txx, .tpp and .tpl for template definitions. Those are either included in the headers providing the definition, or manually in the contexts where they are needed.

Compilers and tools usually don't care about what extensions are used, but using an extension that they associate with C++ prevents the need to track out how to configure them so they correctly recognize the language used.

2017 edit: the experimental module support of Visual Studio recognize .ixx as a default extension for module interfaces, clang++ is recognizing .c++m, .cppm and .cxxm for the same purpose.

I use ".hpp" for C++ headers and ".h" for C language headers. The ".hpp" reminds me that the file contains statements for the C++ language which are not valid for the C language, such as "class" declarations.