On Linux and Unix emacs is either already installed on your machine, or available via your software manager. It consists of several interlocking components though, and it's possible to install only the console version, missing the GUI (graphical) version. There's also xemacs, which for a time was far ahead of emacs in usability but has since fallen somewhat out of favor. Its menus make a lot more sense than emacs', in my opinion, but it's no longer very good looking, and the instructions here won't fully apply.
Mac OSX is Unix under the hood, so it runs emacs natively, and the console version, not the GUI version, is even provided free with new Macs. Open a new Terminal window (look in your applications folder) and enter the command emacs at the prompt, and emacs will start. There are other options as well. Enrico Franconi3 has developed a fully carbonized version of emacs which provides the best of all possible unions between emacs and the Mac OSX operating system. The Mac menubar overhead takes the emacs menu, the program interacts perfectly with the Mac clipboard, and you can launch it directly from the OSX Dock. Moreover, the Apple Command key (``flower'') serves as ``Alt'' (shown as M- in this document), which is far more convenient than reaching for the Escape key as you have to do otherwise.
There's also Aquamacs, a version of emacs that tries to reach a compromise with Apple's desktop, using Macintosh native file dialogs and more. It's comfortable software in a lot of ways, and the first application I reach for on Mac OSX when I want emacs.
For Windows, There is a native version of emacs available from GNU's Savannah server.4