When the kernel finishes loading, it starts a program called init. This program remains running until the system is shut down. It is always assigned process ID 1, as you can see:

$ ps --pid 1
  PID TTY          TIME CMD
    1 ?        00:00:04 init.system

The init program boots the rest of your distribution by running a series of scripts. These scripts typically live in /etc/rc.d/init.d or /etc/init.d, and they perform services such as setting the system's hostname, checking the filesystem for errors, mounting additional filesystems, enabling networking, starting print services, etc. When the scripts complete, init starts a program called getty which displays the login prompt, and you're good to go!