For example, what would it mean to say:
cd ~/mydirectory
~ is a shortcut for the home directory, so
cd ~/mydirectory
Will move to mydirectory if there is such a directory in your home folder, otherwise it will throw a directory does not exist error.
Example:

Your home directory is located at /home/<username>.
Trivia: This is why the default bash prompt has a tilde in it like this user@host:~$. The tilde indicates that you are currently in your home directory. If you move somewhere else, for example into /etc it will show user@host:/etc$.
It's shorthand for your home directory. For example, my desktop directory is at
/home/marc/Desktop
or
~/Desktop
for short.
$HOMEinstead, for clarity and because some programs don't expand a tilde. – kiri Feb 06 '14 at 20:59