The devastating undead plague began in Northrend after the Second War. There, from the depths of the Frozen Throne, the Lich King Ner'zhul afflicted a remote human village through his will alone: a morbid test meant to gauge the plague's effectiveness. The infected villagers died, and when their zombified corpses rose soon after, they had become lumbering, mindless servants of Ner'zhul.
The experiment was successful, but the Lich King was interested in nothing less than perfection. He contaminated every human inhabitant of Northrend, binding them to his icy will even as he continued to fine-tune his infernal disease.

Infested Plaguelands
Through the course of his experimentations, Ner'zhul insured that the affliction would specifically target humans for "undeath". Though non-human races and creatures (and even the land itself) were susceptible to the plague, it was humanity in particular that Ner'zhul meant to scour from the world. As a result, infected flora and fauna reacted differently--diseased and decaying, but not truly undead, and not under the thrall of the Lich King.

Hence, while undead representatives certainly do exist among the ranks of the non-human races, these particular agents are examples of undead created through necromancy rather than the plague.

Some strange
experiments
Once Ner'zhul's adjustments were complete, his mind reached out to Dalaran, to the disgruntled human, Kel'Thuzad. The archmage answered the call, trekking through the arctic wastes of Northrend to eventually climb the steps of the Frozen Throne*. There he pledged to act as the Lich King's lieutenant in exchange for immortality and untold power. He was then given cauldrons of concentrated plague to spread throughout the lands of Lordaeron via his acolytes in the Cult of the Damned, and soon dead villagers throughout the realm began to rise and walk again, marching against the living in obedient servitude to their new master.

And thus, the Scourge was born into an unsuspecting world.

*Also, read the Road to Damnation short story.