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I understand it in careers "Is to move up to a higher position".
Advancement of management or supervisory responsibilities in an organization with an accompanying increase in compensation. An individual receives a vertical promotion from a department manager to the position of vice-president, including an increase in responsibilities and compensation.
In terms of careers this refers to the upward movement in a career from a lower level job to a higher level one with more responsibility, rewards and authority. An example of such would be a promotion from being a supervisor to a manager of a particular section of the company say Operations.
Based on the above, this can also apply to the Education system, a situation whereby a student is promoted to the next level regardless of having learnt the material from the lower class or not usually because they are more intelligent for the class therefore it renders it necessary to push them higher up the ladder.