Although the King (or Regent, in his stead) has absolute authority in Amber, no one else does.
All titled, landed Lords of Amber — the heads of the Houses — have legal authority on their own lands, over residents on those lands, save that they still cannot break any of the Crown's laws. Basically, the Lord has the right of low justice over his own people on his own lands, including dealing with civil matters such as contract disputes. Much of the time, this is actually handled by the Crown's Magistrate, as many of the Lords don't want to spend their time dealing with petty matters. Note that House-owned properties in Amber City are not considered part of a House's lands; they are considered part of Amber City, and subject to the city's laws and justice.
However, the bulk of the population of Amber is concentrated in Amber City, which is a giant melting pot, and its cosmopolitan nature and the relative weight of money (which while less important than birth, is still very important) make its society less clearly feudal. Moreover, foreigners are treated essentially identically to Amber's own citizens of similar social class, with the accordant careful dance of politics for those of high rank or exceptional wealth.
Those appointed by the Crown into particular positions carry the weight of the Crown's authority when making pronouncements related to their jobs. Similarly, their reasonable, job-related requests carry that authority. If one doesn't like the pronouncement or doesn't want to comply with the request, it can be appealed to the Crown, but the Crown has historically had little patience for those interfering with the hands that carry out its will. Protesting to the Crown costs one political capital, and one had better hope that the protest will be judged by the Crown as valid.
Beyond that (and military rank in military situations, of course), people generally have no legal authority to order each other around. Legally speaking, a Lord can't just order the common man in the street to do something for him, although in practice this does happen ("you there, bring my luggage in!") — but it's consider appropriate to toss the fellow some coins for the work, and it's part of "you'll do it if you know what's good for you", as opposed to actual legal authority. This is even true for the Princes. It does apply up the ranks, as well — one of the Lords can refuse a request from one of the Princes, for instance, but it has potential political and social ramifications.
Legality aside, there is tremendous social pressure to respect one's betters. In the case of the lower classes, retribution for a lack of respect can be instant; for instance a scruffy commoner who insults a Lord might very well be grabbed by that Lord's companions, taken out back, and beaten soundly, with the justice system turning a blind eye to the assault. Among the upper classes, the situation is more complicated. If a lower-ranking Lord insults a higher-ranking Lord, that's a matter of honor, something that might be settled by a duel or the like. If a wealthy merchant insults a Lord, that's probably something to be seethed over and settled with some manner of quiet revenge.
