How do I compile a Linux kernel module?

You have to write a Makefile. The Makefile is a standard one and depends on the kernel version. Do not try to make your own Makefile, use the Linux kernel infrastructure, as explained below.

Standard Makefile for 4.0 and later kernels

obj-m := name_of_module.o

clean-files := *.o *.ko *.mod.c

and then use:

cd path/to/external_module
make -C /path/to/kernel/source M=$PWD

Standard Makefile for 2.6 kernels

obj-m := name_of_module.o

all:
        make -C /lib/modules/$(shell uname -r)/build M=$(PWD) modules

clean:
        make -C /lib/modules/$(shell uname -r)/build M=$(PWD) clean

For more informations, see Linux Device Drivers, Building and Running Modules, and Driver porting: compiling external modules.

Standard Makefile for 2.4 (and <=) kernels

TARGET := module_name
INCLUDE := -I/lib/modules/`uname -r`/build/include
CFLAGS := -O2 -Wall -DMODULE -D__KERNEL__ -DLINUX
CC := gcc

${TARGET}.o: ${TARGET}.c
        $(CC) $(CFLAGS) ${INCLUDE} -c ${TARGET}.c

Once the module is compiled, you can load it with the following command :

insmod module.[ko|o]