The Taint is a form of dark magic which can infect anything living, which is one of the reasons the Dwarves developed Golems in the first place as they appear to be immune to it.

The Taint is spread by the 'bodily fluids' of darkspawn, usually their blood, and corrupts and mutates the flesh. In most cases this quickly kills anyone who's infected, since they develop internal tumours which disrupt their biological functions, but in some cases it acts as a neurotoxin, infecting the brain and causing madness long before the rest of their body succumbs. These people are referred to as Ghouls in Thedas.

In rare cases the infected person's body can adapt to the Taint, becoming a Grey Warden, but over time they also mutate into freaks and madmen, usually choosing to head alone into the Deep Roads to end their days with honour once they start to hear the Calling. Those that survive their expeditions become dark and perverted versions of their former selves, retaining their own sense of self and intellect, even though their minds are often twisted and insane.

The Architect recently discovered a way to stabilise those infected with the Taint, turning them into Awakened who were able to retain their sanity despite the ravages mutating their bodies.

The most stable darkspawn are those born from a broodmother, who herself is a female from any race who is brutally raped, tortured and forced to eat Tainted flesh for many days before becoming the mutated mound of flesh which gives birth to the various types of darkspawn. For instance, dwarven females give birth to Genlocks, human female birth Hurlocks and Kossith females give birth to Ogres (luckily there are very few Kossith living in most of Thedas, whether Tal-Vashoth, Vashoth or Qunari, and so Ogres tend also to be rare.)

Based on [@WildCobra]'s description and backstory for Karras he'd actually be an Awakened Ghoul rather than a true darkspawn, but I doubt that many people in Thedas would worry about the distinction, since pretty much anyone who looks darkspawn-y would be referred to as darkspawn no matter what their actual condition may be.