The world of Avatar: The Last Airbender proved immediately engaging from the start of the series, immersing fans so much that it's hard to sometimes appreciate just how little of it viewers saw in the original series. Although Aang and his friends traveled all around the world, each episode saw only a slice of each place they went to. This leaves plenty of space for confusion, which has some fans claiming that Katara is technically a princess.
Born the daughter of a highly-respected leader of the Southern Water Tribe, the claims that Katara is a princess are not totally baseless. However, looking over the series, both in what the original show reveals and other information gained about the franchise since, the case is pretty clear that Katara is not a princess at all.
At the start of the series, Katara and Sokka live with their Gran Gran, having lost their mother to the Hundred Year War during their childhood and missing their father since he left to aid the war efforts. Going by the original series itself, it's understandable if viewers think the Southern Water Tribe consists solely of Katara, Sokka, Gran Gran and the small smattering of other people seen to inhabit their village. When their patriarch Hakoda appears later in the series and serves a leadership role, it's also easy to mistake him as the leader of the entire Southern Water Tribe.
However, this is not the case. Just one of several tribes are seen during the original series, and the Southern Water Tribe is actually significantly larger than the initial two episodes of Avatar lead viewers to believe. Similarly, Hakoda does not occupy a position of political power supreme to any of his fellow countrymen; he is a minor chieftain whose primary role was leading the war parties that harried Fire Nation forces during the latter days of the war. Hakoda served as part of the Southern Tribe's tribal confederation, sharing power equally with several other village leaders -- far different from a king.
While Hakoda did become a Head Chieftain following the war, the position was one he was elected to rather than owed by birthright. Notice the language of chiefs and chieftains is far different from noble titles like king and princess, and when compared to the Fire Nation royal family, the discrepancy only becomes all the more apparent. Princess Azula owed her status to a long line of generations of rulers whose power was autocratic, largely uncontested, and far from the democratic and respect-based systems of the Southern Water Tribe.
While such considerations may seem all the more respectable from a modern standpoint, they do make Azula's references to Katara as a "peasant" far more understandable. From Azula's point of view, Katara and Sokka hail from a backwater country with a completely alien system. Neither of them shares the proud lineage by which Azula defines so much of herself. That may be exactly why Azula underestimates Katara so thoroughly in the finale, foolishly opening herself up for an attack that allows Katara to win their duel.
As culturally diverse as the influences are behind the world of Avatar, it would be a shame to reduce the spectrum of political positions to European titles like "king" or "princess" when they don't really fit. Katara never really needed a title to be one of the strongest figures on the planet. "Princess" would have been okay, but as possibly the greatest waterbending master in the world, that seems like plenty.