There are many Assembly compilers out there. nasm is one compiler that is available across platforms and can be used to program x86 processors.

Use interrupt 0x80 to make Linux system calls like reading and writing to file descriptors and sockets. For a list of system calls in Linux, refer to https://man7.org/linux/man-pages/man2/syscalls.2.html

Here is an example of compiling with nasm:

nasm -f elf64 file.asm 
# or for 32-bit
nasm -f elf file.asm

If you need to link against something you can use:

ln -d -o outfile file.o

Here is a Hello World example:

hello.asm

SECTION .DATA
	hello:     db 'Hello world!',10
	helloLen:  equ $-hello
 
SECTION .TEXT
	GLOBAL _start 
 
_start:
	mov eax,4            ; 'write' system call = 4
	mov ebx,2            ; file descriptor 1 = STDOUT
	mov ecx,hello        ; string to write
	mov edx,helloLen     ; length of string to write
	int 80h              ; call the kernel
 
	; Terminate program
	mov eax,1            ; 'exit' system call
	mov ebx,0            ; exit with error code 0
	int 80h              ; call the kernel

Use interrupt 0x21 to make DOS system calls like reading and writing files. For a list of DOS system calls, refer to https://en.wikipedia.org/wiki/DOS_API

When interacting with the BIOS, you use a different system call for each function. BIOS will have less system calls available than an kernel like DOS or Linux, but it gives you the tools you need to build an operating system. For example, BIOS will let you change the video mode, get input from keyboard, write text to screen, and draw pixels on the screen.

For a list of BIOS system calls, refer to https://en.wikipedia.org/wiki/BIOS_interrupt_call.

You can write and compile libraries that can be linked against by other programs.

This example shows how to create a function called print_hello() that can be used from other Assembly or C programs.

static_lib.asm

; Compile this program using
; nasm -f elf64 static_lib.asm 
; gcc myprogram.c static_lib.o
; ./a.out
 
SECTION .DATA
	hello:     db 'Hello world!',10
	helloLen:  equ $-hello
 
SECTION .TEXT
	GLOBAL print_hello
 
print_hello:
	mov eax,4            ; 'write' system call = 4
	mov ebx,2            ; file descriptor 1 = STDOUT
	mov ecx,hello        ; string to write
	mov edx,helloLen     ; length of string to write
	int 80h              ; call the kernel
 
	; Terminate program
	mov eax,1            ; 'exit' system call
	mov ebx,0            ; exit with error code 0
	int 80h              ; call the kernel