CFLAGS, CCFLAGS, CXXFLAGS - what exactly do these variables control?

According to the GNU make manual:

CFLAGS: Extra flags to give to the C compiler.
CXXFLAGS: Extra flags to give to the C++ compiler.
CPPFLAGS: Extra flags to give to the C preprocessor and programs that use it (the C and Fortran compilers).

src: https://www.gnu.org/software/make/manual/make.html#index-CFLAGS
note: PP stands for PreProcessor (and not Plus Plus), i.e.

CPP: Program for running the C preprocessor, with results to standard output; default ‘$(CC) -E’.

These variables are used by the implicit rules of make

Compiling C programs
n.o is made automatically from n.c with a recipe of the form
‘$(CC) $(CPPFLAGS) $(CFLAGS) -c’.

Compiling C++ programs
n.o is made automatically from n.cc, n.cpp, or n.C with a recipe of the form
‘$(CXX) $(CPPFLAGS) $(CXXFLAGS) -c’.
We encourage you to use the suffix ‘.cc’ for C++ source files instead of ‘.C’.

src: https://www.gnu.org/software/make/manual/make.html#Catalogue-of-Rules

As you noticed, these are Makefile {macros or variables}, not compiler options. They implement a set of conventions. (Macros is an old name for them, still used by some. GNU make doc calls them variables.)

The only reason that the names matter is the default make rules, visible via make -p, which use some of them.

If you write all your own rules, you get to pick all your own macro names.

In a vanilla gnu make, there's no such thing as CCFLAGS. There are CFLAGS, CPPFLAGS, and CXXFLAGS. CFLAGS for the C compiler, CXXFLAGS for C++, and CPPFLAGS for both.

Why is CPPFLAGS in both? Conventionally, it's the home of preprocessor flags (-D, -U) and both c and c++ use them. Now, the assumption that everyone wants the same define environment for c and c++ is perhaps questionable, but traditional.

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P.S. As noted by James Moore, some projects use CPPFLAGS for flags to the C++ compiler, not flags to the C preprocessor. The Android NDK, for one huge example.