libstdc++ is the GNU c++ standard library implementation.

libc++ is the LLVM/clang c++ standard library implementation.

Even when compiling with clang, libstdc++ (gnu) is often used (on Linux).

A main reason libc++ (clang) exists is that libstdc++ (gnu) is GPL and so Apple can't ship it, so you can think of libc++ as the non-GPL libstdc++.

To find which library is being used you could run

Copy $ /sbin/ldconfig -p | grep stdc++
    libstdc++.so.6 (libc6) => /usr/lib/libstdc++.so.6

The list of compatible versions for libstdc++ version 3.4.0 and above is provided by

Copy $ strings /usr/lib/libstdc++.so.6 | grep LIBCXX
 GLIBCXX_3.4
 GLIBCXX_3.4.1
 GLIBCXX_3.4.2
 ...

For earlier versions the symbol GLIBCPP is defined.

The date stamp of the library is defined in a macro __GLIBCXX__ or __GLIBCPP__ depending on the version:

Copy// libdatestamp.cxx
#include <cstdio>

int main(int argc, char* argv[]){
#ifdef __GLIBCPP__
    std::printf("GLIBCPP: %d\n",__GLIBCPP__);
#endif
#ifdef __GLIBCXX__
    std::printf("GLIBCXX: %d\n",__GLIBCXX__);
#endif
   return 0;
}

$ g++ libdatestamp.cxx -o libdatestamp
$ ./libdatestamp
GLIBCXX: 20101208

The table of datestamps of libstdc++ versions is listed in the documentation:

What is libstdc++11?

What is libstdc++.so.6? Is this just a library that gcc ships with?

It’s GCC’s implementation of the C++ Standard Library.

What are all the different GLIBCXX_*? Are these also libraries?

These are version symbols, used to allow the library to remain backward-compatible while still being able to change its API. See What do the multiple GLIBC versions mean in the output of ldd? for details.

Question is why do I have different versions of these libraries in different folders?

The copy in /usr/lib64 is your distribution’s version; the copy in /usr/local/lib64 was installed by something else.

When running gcc how do I tell it to use the 6.0.20 library in /usr/local/lib64 instead of the older one?

You can try building with -L/usr/local/lib64, and running with LD_LIBRARY_PATH=/usr/local/lib64 (if necessary).

The LIBRARY_PATH environment variable is pretty standard. It is known to majority of compilers.

You should also use C_INCLUDE_PATH and/or CPLUS_INCLUDE_PATH. These two a more gcc specific (other compilers prefer INCLUDE without language separation).

You can also ignore the environment variables completely and specify the correct libstdc++ directly in the command line.

g++ main.cpp /software/gcc10/.../libstdc++.a

There isn't actually a package called libstdc++ (possibly you are trying to install libstdc++6?)

When apt doesn't find an exact match to a package name, it treats the given string as a regular expression. In this case the ++ causes it to try to install any package whose name matches libstd followed by one or more c - many of which conflict, as you can see.

Unfortunately I have never managed to figure out how to turn off this behaviour (nothing I have tried in the past using escape characters has seemed to work) - the only solution I know is to be careful to give an exact package name where any regex characters might be misinterpreted, and to make sure the package catalog is up-to-date i.e.

sudo apt-get update
sudo apt-get install libstdc++6