Watch these videos. The first explains the Brønsted-Lowry definition of acids and bases. The second explains how auto-ionization of water molecules happens. Water is an amphiprotic compound: it can act as either an acid or a base. Hydrogen ions do not exist in water on their own but immediately get grabbed by water molecules to form hydronium ions. The third explains the relationship between conjugate acids and bases. A conjugate pair is always one acid and one base.
Source: Khan Academy
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