Avatar: The Last Airbender is a fantasy action series that introduces the bending combat system, wherein users can manipulate the four elements in battle. The hero, Avatar Aang, can bend all four, while his teammate Katara is a dedicated Waterbender and his friend Toph Beifong is an expert Earthbender.
Katara and Toph have always been on the same side, but they've also clashed a few times. Their fight in the Book Three episode "The Runaway" was interrupted by Aang, but if Katara and Toph were to really fight to see who's the better bender, how would that duel turn out?
Katara: The Southern-Style Bender
Katara was born and raised in the Southern Water Tribe with her non-bender brother Sokka. At this time, no other benders could be found in the village. Katara had no proper teacher, so she was self-taught, and she was greatly motivated to protect her tribe with her unique abilities. When Aang came along, Katara got a chance to explore the world and practice her bending some more, and she even stole and learned from a rare waterbending scroll (to be fair, it was already stolen by pirates). Katara's bending got another boost when she learned alongside Aang from master Pakku in the Northern Water Tribe. Later still, Katara met the rogue Waterbender Hama and learned to not only bend the water in plants, animals and even the blood in other people. Katara vowed to never use bloodbending again if she could help it.
Katara's fighting style began as improvised versions of the original Southern style, but she learned many elements of the Northern style from Pakku and has improvised many more techniques. She can create a huge water octopus to defend herself or grab a foe with watery tentacles, or summon a surge of water to sweep away her foes with one blow. Katara is also comfortable with ice, being able to form icy ramps to slide around, barriers to block an enemy's attack and sharp icicles to use as projectiles. She's fond of using the water whip technique.
Toph Beifong: Learning From The Badger-Moles
Unlike most benders, Toph Beifong didn't learn bending from a person, but instead from the badger-moles, the original Earthbenders. Like Toph, the badger-moles are blind and compensate with their sensitive paws that can detect vibrations in the Earth itself. After spending years with the badger-moles, Toph developed a highly perceptive earthbending style. She can sense vibrations with her feet, and locate any person or object nearby. This allows Toph to "see" people and animals even through walls or underground, and she once heard dozens of captives crying for release in a nearby mountain using this method.
Such a sensitive fighting style gives Toph a serious edge in bender battles because she can feel every minute movement in her opponent's body and thus read them. Toph easily read and countered The Boulder's strike using this method. Toph always strikes second, and always strikes last. Toph also understands that defeating an opponent means dealing rapid blows to knock them off-balance, then delivering a final blow when they are most vulnerable.
For most of Avatar: The Last Airbender, Toph used fairly conventional earthbending techniques, such as creating barriers, throwing boulders, knocking foes aside with earth pillars and even bending metal. In the Season 2 finale, she invented metalbending after realizing that metal is merely an extension of the Earth itself, and she'd go on to master and train others in this style.
The Bending Duel: Toph Vs Katara
Katara knows that Toph cannot easily see with ice, and Toph knows that Katara will be thinking along these lines. In terms of sheer firepower, Toph has only a small edge on Katara, so the real deciding factor is whether Toph can maintain her earth-based vision or not. Once the duel starts, Katara must summon some water and harden it into ice, but doing that means performing distinct bending moves, and Toph can sense that. Toph can act fast and sink Katara into the ground, and ideally trap her arms. If Toph can trap Katara's arms before enough ice forms, then Toph can maintain her vision and deal the finishing blow. Once Katara is securely locked in hardened earth, she must give up. Not even bloodbending can get her out of this one.