System calls allow programs to interface with the kernel and perform tasks like input/output. In assembly, system calls are made by putting the system call number in the EAX register, arguments in registers like EBX and ECX, and calling the interrupt 80h. Common system calls include sys_write to output to the screen and sys_exit to terminate a program. The example code demonstrates using sys_write to prompt for user input, sys_read to get input, and sys_write again to display it before exiting with sys_exit.