A base is the partner of an acid: it picks up H⁺. To hold on to an H⁺, a base needs a pair of electrons to make a bond with.
A lone pair is a pair of outer electrons that an atom keeps to itself. Ammonia's nitrogen has one. An amine is ammonia with carbon groups in place of hydrogens, and its nitrogen keeps that lone pair. The lone pair reaches out to an H⁺ and becomes a new N–H bond. The amine is now an ion with a positive charge.
In water, a base takes H⁺ from water molecules and leaves hydroxide ions, OH⁻, behind. A solution with more OH⁻ than H⁺ has a pH above 7. A base that dissolves in water is called an alkali.
Lemon with fish
Trimethylamine is the amine behind the smell of fish that is going off. It drifts out of the fish into the air and up your nose. A squeeze of lemon juice gives it an H⁺ from citric acid. The ion it becomes carries a charge, so it stays dissolved in the juice, and the smell fades.
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Baking soda
Baking soda, also sold as bicarbonate of soda, is sodium hydrogencarbonate, NaHCO₃. It is made of sodium ions, Na⁺, and hydrogencarbonate ions, HCO₃⁻. Each hydrogencarbonate ion has an oxygen with a negative charge, ready to pick up an H⁺. Dissolved in water, baking soda gives a pH of about 8.3: gently basic.