In Single-User Mode, the operating system is not running any of the normal services. It cannot be logged into over the network, nor does it require you to log in as root to do administrative tasks. It is a bit like Recovery mode on Windows, but significantly more powerful.
You can explicitly choose to boot into single-user mode from the Boot Menu. Alternatively, if some parts of the normal boot process fail, such as Filesystem checks, the system will switch to single-user mode automatically to allow you to resolve the problem (the Windows equivalent would be to Blue Screen at which point you can do nothing).
You will see the following upon entering single-user mode:
Enter pathname of shell or RETURN for /bin/sh:
Simply hit RETURN and you will see a # prompt which means you are logged in as root and can run any normal commands. It is very likely that no filesystems will be mounted (the root filesystem will be read-only) at this stage.
To get out of single-user mode, you can either run a command such as
halt or type Ctrl-D at the prompt to boot as normal into multi-user mode.