If my Mac was to go dead or get stolen, how does my back up hard drive with Time machine kick in?

In other words, my Mac is gone. I go to the Apple Store and purchase a new Mac. Now can I just hook up my external back-up hard drive to my new Mac and start as if I had my stolen computer? What actually happens? Has this ever happened to you? The reason I ask is because I always back my whole computer up with Time Machine and my backup hard drive, but then I wonder how the restore process actually works if I got a new Mac or if my current Mac just dies out and needs to be installed from a blank slate. Can you literally use Time Machine to install all your files, preferences, programs, etc. as if it never happened?