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What happens when a substance ionizes?

Right now in Chemistry we're doing acids and bases. I don't really understand what it means when something ionizes. For example, when water ionizes into hydroxide and hydrogen ions, what is happening? Also, what does it mean when a strong acid completely ionizes in water, while a weak acid only partially ionizes in water? Are they breaking down? Finally, on quizzes, our teacher has us write a chemical equation of a substance ionizing. When something ionizes, does it always go into equilibrium?

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