Decide which is going to be your default platform - say LINUX.
LIBS_PATH = -L/usr/norton/lib
INC_PATH = -I/usr/norton/include
LIBS = -lntr
PLATFORM = -DLINUX
CXX = g++
app: *.cpp *.h Makefile
${CXX} ${CFLAGS} ${PLATFORM} ${INC_PATH} *.cpp -o $@ ${LIBS_PATH} ${LIBS}
You can use round brackets in place of braces. This uses a macro for the C++ compiler, allows you to add other flags via CFLAGS (though that is also usually set by 'make'), and adds a platform, the include path, the library path and the actual library to the compile line.
Note that your rule enforces a complete recompilation of everything every time anything changes. This is 'safe' but not necessarily efficient. Note that wild-cards are dangerous too - more so for the source than the headers. You may include that backup copy of a file in the build (old-reader.cpp - you only wanted reader.cpp in there really). More conventionally, you list the object files needed for the program so that each object file can be individually rebuilt when needed, and the results linked together. If you get your dependencies correct (a moderately big 'if'), then there's no problem. If you get them wrong, you can end up with inconsistent programs.
However, if the difference is between a 5 second recompile and a 5 minute recompile, you should probably take the 5 minute recompilation (as shown) and answer another SO question while waiting.
To compile on Linux (64-bit):
make CFLAGS="-m64"
To compile on Linux (32-bit):
make CFLAGS="-m32"
To compile on Windows 64:
make PLATFORM=-DWIN64
To compile on Windows 32:
make PLATFORM=-DWIN32
Etc.