I\'m only new to using SSH, but when I log in I end up in the directory ~, which in general is the same directory when FTPing in. I can still go to /, but I don\'t know what
~ expands to your home directory, as has been pointed out, but I think it's worth noting that isn't a feature of ssh itself.
ssh (among many other wonderful features!) lets you establish a remote shell, and this shell can provided by many different pieces of software.
On a *nix system, your account will be associated with a particular shell, GNU bash is a popular choice. And it so happens than in bash, and most other POSIX compliant shells, the tilde character expands as a shortcut to your home directory.
/ is the root of the file system ~/ or ~ is the root of your profile, ~/ is always /home/username
And home, in relation to /, isn't necessarily always in the same place. That's why the ~ shortcut is so useful. The path to home should be in $HOME. Try typing
echo $HOME
~ is your home directory, yes. Which is very nice since your home directory is not always where you think it should be (/home/).
Also, fun fact: You can use "cd ~myuser" to get to the home directory of the user "myuser".
Different shells may or may not handle this differently, but Johnathan got the closest without coming out and saying it. The shell expands "~" to whatever's stored in the $HOME environment variable. The shell expands ~username to whatever's listed in the shell field of /etc/passwd for the given username. If you don't override it, the shell (or ssh, depending on the implementation) will set $HOME to be the home field from /etc/passwd, so they're both the same (assuming you're "username") until you change one.
As to why you see a ~ in ssh...
The prompt says "~" is your current directory most likely because you're using Bash as your shell, and the value of $PS1 (the prompt string you see - it's set in /etc/profile or /etc/profile.d/*, more than likely) contains a \w or a \W somewhere. The \w string in the prompt shows the current directory, and collapses to a "~" when you're in the directory specified by $HOME. Here's a little demo starting in my homedir - note how the "\w" gets replaced with either the current directory or with a ~, based on what the value of HOME is set to. Also note that the trailing slash doesn't work - HOME can't end with a slash for this to work. :)
danny@machine ~ > export PS1='\w > ' # change my prompt (effective on next line)
~ > cd /tmp # move to /tmp to demonstrate
/tmp > export HOME=/tmp/ # set HOME to include trailing /
/tmp > export HOME=/tmp # try again without trailing /
~ > # notice that this works
~ > cd /home/danny # back to homedir
/home/danny > export HOME=/home/danny # see how it's /home/danny, not ~
~ > export PS1='danny@machine \w > ' # after resetting $HOME, it should work ok
danny@machine ~ > # hooray!