I'll give this a shot. A pure turbojet engine is one that doesn't have any bypass air. An example would be the engines on an old DC-9. A Turbofan is a turbojet engine that has a fan strapped to the front of it. Some of the air from the fan is ducted around the center turbojet section.
In addition to that you have low-bypass turbofan engines and high-bypass turbofan engines. An example of a low-bypass fan engine is an MD
-80's JT8D. There is not a lot of air that bypasses the actual turbojet section. A good visible example of a high-bypass fan engine is the GE
fan on the 747-400. If you look in pictures you can see a gap between the jet section and the engine cowling - this is where the bypass air flows through. Hope this helps.