Soot built up in the exhaust which is burnt off when a load of hot gases are thrown down the exhaust. Usually happens when the engine is driven gently for a while then hard acceleration.
Regeneration shouldn't produce a lot of smoke though. It should happen at very high temperatures, which lead to relatively clean combustion of the stored soot - it's not simply belching out of the back.
As mentioned before it's basically just that diesels run a much richer air-to-fuel ratio than petrol engines and also the chemistry of diesel combustion lends itself more to particulate production.
Modern engines have got a lot better, principally because virtually all now have diesel particulate filters (the matter trap that collects the soot mentioned above) and this acts like a giant tea-strainer in the exhaust, physically trapping the soot. You ideally want a steady state with a reasonable amount of gas flow to regen, so the ECU recognises this state - ideally on a long motorway cruise - and post-injects to dramatically increase the temperature in the exhaust to about 1500 deg C mid-brick.