What is Communism?

Communism, as an economic process, is a society or social formation in which the communist class process prevails. The communist class process is one of many different ways surplus labor is appropriated and distributed.  The communist class process is distinguished by the collective appropriation of surplus labor by the same workers who performed the surplus labor.  A prominent contemporary example of communism is the kibbutz system in Israel.  Communism was also common in pre-colonial indigenous societies in Africa and North America.  This economic notion of communism is to be contrasted to the more recent political use of the term (in the so-called West) to refer to a political structure within which a communist party holds a monopoly over state powers. Note that in such societies, the term that is commonly used is "socialism" not communism to refer to societies ruled by a communist party. for a more elaborated discussion of this topic, see Gabriel, Chinese Capitalism and the Modernist Vision.
Communism is often confused with Stalinism: the social system (which included state capitalism) that developed under the dictatorship of Joseph Stalin and which was grounded in certain social processes developed during the period of war communism in the Soviet Union during the Civil War against Western supported White Russian forces.  Communism is based on communal production and appropriation of surplus labor (in direct labor, product, or exchange value forms) and, as such, does not require equal distribution (as is often asserted).  As is the case in the kibbutz system, communal appropriation and distribution simply requires some democratic form of distribution of the collectively appropriated surplus. 
I have reconstructed the fundamental
communal nature of these
(pre-Columbian) social formations . . .
In these communal societies
means of production are held in common,
and access to strategic
factors of production is guaranteed.
Surplus labor
-- required for care of the sick and infirm,
replacement of strategic
factors of production,
and maintenance
of socioceremonial life,
among other activities --
is collectively appropriated.
Primary producers participate in decision making
about the amounts of surplus produced,
its form (products or labor service),
and its conditions of production.

---Dean Saitta, in Re/presenting Class (2001)