A "plan" is technically nothing more than a set of government-sanctioned goals regarding economic development.

In Western (or Western-aligned) countries where such plans were implemented, state authorities didn't mean to question the validity of a market-based economy but they did see it as necessary for the government to get significantly involved and steer markets into certain directions relevant to the "national interest" — such as fostering rapid recovery and modernization in the aftermath of WWII, for instance. In that context, "planning" is conceived of as a technocratic mitigation of the market's insufficiencies. In France, a "General Commissariat for the Plan" was created in 1946 (and existed until 2006).

The USSR, the PRC and their satellites on the other hand operated as command economies. The highly centralized state formulated fairly strict productions quotas and the chain of command was expected to enforce them all the way down, because that was the only way to ensure they would be met. The idea was to do away with markets altogether, not just to control or direct them. In that context, "planning" is touted as a superior economic model around which the whole economy ought to be structured. The PRC no longer strictly adheres to that model, which is why the Chinese government has been speaking of "guidelines" rather than "plans" since the 11th five-year plan.

Certainly, you can now see why "plans" are more essential to economic models that rely on them entirely as opposed to economic models that see them as merely corrective. You would be right however in pointing out the many similarities in both types of planning. The lack of differentiation in many departments is what led many left-wing theorists to define the USSR or PRC as "state-capitalist" entities.

Planning as a general trend emerged in the early 20th century, as a reaction to the crisis of liberalism. Liberalism was seen by many as unjust, unstable and untrustworthy. They thought capitalism needed to be saved from itself, or replaced by a better system. "Planists" of many persuasions (from New Deal democrats to Bolsheviks, from fascist regimes to emergent third-world powers) argued that a better model that was fair, efficient and reliable was possible. And eventually, varying degrees of planism came to characterize most of post-WWII national economies... until around the '80s.

This model itself came into crisis (at least in developed countries) at that time, mainly because it arguably belongs to an era of industrialization, strong nation-states and bureaucratic machines (i.e. economies being centered around state entities) that started being displaced by globalization. But that's another subject entirely!