A login shell is the first process that executes under your user ID when you log in for an interactive session. The login process tells the shell to behave as a login shell with a convention: passing argument 0, which is normally the name of the shell executable, with a – character prepended (e.g. -bash whereas it would normally be bash. Login shells typically read a file that does things like setting environment variables: /etc/profile and ~/.profile for the traditional Bourne shell, ~/.bash_profile additionally for bash†, /etc/zprofile and ~/.zprofile for zsh†, /etc/csh.login and ~/.login for csh, etc.

When you log in on a text console, or through SSH, or with su -, you get an interactive login shell. When you log in in graphical mode (on an X display manager), you don’t get a login shell, instead you get a session manager or a window manager.

It’s rare to run a non-interactive login shell, but some X settings do that when you log in with a display manager, so as to arrange to read the profile files. Other settings (this depends on the distribution and on the display manager) read /etc/profile and ~/.profile explicitly, or don’t read them. Another way to get a non-interactive login shell is to log in remotely with a command passed through standard input which is not a terminal, e.g. ssh example.com <my-script-which-is-stored-locally (as opposed to ssh example.com my-script-which-is-on-the-remote-machine, which runs a non-interactive, non-login shell).

When you start a shell in a terminal in an existing session (screen, X terminal, Emacs terminal buffer, a shell inside another, etc.), you get an interactive, non-login shell. That shell might read a shell configuration file (~/.bashrc for bash invoked as bash, /etc/zshrc and ~/.zshrc for zsh, /etc/csh.cshrc and ~/.cshrc for csh, the file indicated by the ENV variable for POSIX/XSI-compliant shells such as dash, ksh, and bash when invoked as sh, $ENV if set and ~/.mkshrc for mksh, etc.).

When a shell runs a script or a command passed on its command line, it's a non-interactive, non-login shell. Such shells run all the time: it's very common that when a program calls another program, it really runs a tiny script in a shell to invoke that other program. Some shells read a startup file in this case (bash runs the file indicated by the BASH_ENV variable, zsh runs /etc/zshenv and ~/.zshenv), but this is risky: the shell can be invoked in all sorts of contexts, and there's hardly anything you can do that might not break something.

† I'm simplifying a little, see the manual for the gory details.