The [Russian] government would tax the landowners on a regular basis, with the assessment based on how many serfs (or "souls") the landowner had on their records at the time of the collection. These records were determined by census, but censuses in this period were infrequent, far less so than the tax collection, so landowners would often find themselves in the position of paying taxes on serfs that were no longer living, yet were registered on the census to them, thus they were paying on "dead souls."
It is these dead souls, manifested as property, that Chichikov seeks to purchase from people in the villages he visits; he merely tells the prospective sellers that he has a use for them, and that the sellers would be better off anyway, since selling them would relieve the present owners of a needless tax burden…
Chichikov's macabre mission to acquire "dead souls" is actually just another complicated scheme to inflate his social standing (essentially a 19th century Russian version of the ever popular "get rich quick" scheme). He hopes to collect the legal ownership rights to dead serfs as a way of inflating his apparent wealth and power. Once he acquires enough dead souls, he will retire to a large farm and take out an enormous loan against them, finally acquiring the great wealth he desires.
So every time I see another article or an ad about how to acquire more followers on twitter, friends on Facebook, or otherwise collect more "souls" for money, fame, or reputation, I start thinking about Chichikov. He did come to an ignominous end, finally fleeing town. Makes me wonder.