Max Weber:
Max Weber describes legitimacy in terms of the citizens’ willingness to obey the commands of the rulers (“Legitimitätsglaube”). Weber defines three sources of legitimacy: (1) Tradition – the system has been there for a long time and is not questioned, (2) charisma of the leader and (3) legality (trust in the justness of the law).
As a criticism it is often stated that Weber’s concept disregards any normative evaluation, although legitimacy is per definitionem a normative concept. Instead, Weber is said to describe political stability (e.g. Mommsen 1974, Friedrich 1963).

Seymour Martin Lipset:
Seymour Martin Lipset defines legitimacy as the ‘capacity of the system to engender and maintain the belief that the existing political institutions are the most appropriate ones for the society’ (Lipset 1981). Lipset thereby distinguishes between actual performance (‘effectiveness’) and perceived performance.
As one criticism one might argue that Lipset’s definition is not stable over time: A populist and ineffective government might have been considered legitimate by misleading its people during its rule. However, after they realize the ineffectiveness they might consider it illegitimate in retrospective.

Sources:
Friedrich, C.J.  (1963): Man and His Government; an Empirical Theory of Politics. New York
Mommsen, W. (1974): Age of Bureaucracy. Oxford
Lipset, S.M. (1981): Political Man: The Social Bases of Politics. Baltimore