What is the difference between a cold-stream thrust reverser and a conventional one? It seems that all engines emit hot air, due to the combustion process.
Not all air that flows through a turbofan engine undergoes combustion. In a turbofan, part of the air (in some engines, most of the air) is bypassed around the power turbine; that air is simply accelerated by the compressor fan and is vented, cold, out the back of the engine to provide thrust.
A cold-stream thrust reverser simply deflects this cold, bypassed air forward, whereas a "normal" thrust reverser deflects the main engine exhaust.
Wikipedia has some pictures and more description.
Adding images to already existing answers.
Hot flow (primary flow)
These reversers use both air from the combustion in the core (purple) and air from the flow bypassing the core (blue):
This design was used early on turbojets, and is still used on turbofans with a limited bypass ratio.
Cold flow (secondary or bypassed flow)
These ones use only the bypassed flow accelerated by the fan (the large rotor at the front of the engine, here in gray):
Modern turbofan engines have large bypass ratios, e.g. 80% of air is bypassed. It makes sense to use reversers only on this flow. As the flow bypassing the core doesn't take part in the combustion process, it is known as the "cold flow".
(One door is open, the other is closed, this is for illustration only ; doors are always either all open or all closed).
Some engines had the two types together (perhaps the early Pratt & Whitney engine of the McDonnell Douglas C-17, but I'm not sure):
This combination was used to get a powerful reverse thrust.
The nice colorful images have been taken from this site unfortunately only in French: L'avionnaire.