The single most important service in a UNIX system is provided by init. init is started as the first process of every UNIX system, as the last thing the kernel does when it boots. When init starts, it continues the boot process by doing various startup chores (checking and mounting filesystems, starting daemons, etc).
The exact list of things that init does depends on which flavor it is; there are several to choose from. init usually provides the concept of single user mode, in which no one can log in and root uses a shell at the console; the usual mode is called multiuser mode. Some flavors generalize this as run levels; single and multiuser modes are considered to be two run levels, and there can be additional ones as well, for example, to run X on the console.
When the system is running, the two most important tasks of init is to make sure gettys are working (to make sure logins work), that various daemons are running, and to adopt orphan processes (processes whose parent has died; in UNIX all processes must be in a single tree, so orphans must be adopted).
When the system is shut down, it is init that is in charge of killing all other processes, unmounting all filesystems and stopping the processor, along with anything else that it feels like doing.