Which type of organization means that there are few or no levels of managers in between the top people who run the company and the employees?

The type of organization is "flat organization".

Other names that are used for the flat organization are:

horizontal organization or delayering organization.

The structure of flat organization alludes to having a generally modest number of layers in your organization's authoritative graph. This is because of a generally higher traverse of control and moderately less layers of administration.