What is missing from a non-afterburning engine to prohibit the use of afterburning?
The afterburner, of course. 

In simplest sense, the afterburner is just an extension attached to the engine where the fuel is dumped into the exhaust, resulting in an inefficient but huge increase in thrust.



Turbojet with and without afterburner; image from aerospaceweb.org

In the above image, you can see the basic schematic of the afterburner- it basically consists of a long tube placed in between the turbine and the nozzle in which additional fuel is added and burned. The (afterburner) tube is quite simple, consisting of mechanisms to spray fuel into the exhaust and ignite it.

This image shows an actual afterburner, from the F/A -18 (F404 engine). As you can see, its quite simple.



... sailor Ryan Draper ... inspects an F/A-18 afterburner in the jet shop aboard the Nimitz-class aircraft carrier USS George Washington (CVN 73). Image from gizmodo.com