How to run 'emacs' in the terminal emulator without a separate window?

I'm no emacs expert, but this did the trick for me:

emacs -nw

You might prefer starting emacs as emacs --daemon and then using emacsclient -t to edit files. This keeps Emacs loaded in the background, so that you can use Emacs more like vi. Editing sessions will start up instantly, and "quitting" via C-x C-c will just close the current connection. It is very nice.

Although your question has been answered, I would venture to say that this is not the way any regular user would use emacs. Unlike vim, emacs generally has a longer startup time and anyway tends to be running all the time with multiple open buffers. So when you want to do something with emacs in a terminal, you invoke emacsclient -nw (assuming an emacs serverhas been started). Look at How to start to use EmacsClient for details.

If you use bash, you can always add the following to your .bashrc

alias emacs="emacs -nw"

That's saved me a lot of typing (and cursing when I forget to type the switches).