What happens when baking soda and vinegar are combined?

When Baking soda and vinegar are combined, it makes a fizzing reaction when the Acetic acid in the vinegar reacts with Sodium Bicarbonate (the chemical name for baking soda).

The result is some water, Sodium Acetate and Carbon Dioxide gas (the bubbles).

What actually happens is this: the acetic acid (that's what makes vinegar sour) reacts with sodium bicarbonate (a compound that's in baking soda) to form carbonic acid. It's really a double replacement reaction. Carbonic acid is unstable, and it immediately falls apart into carbon dioxide and water (it's a decomposition reaction). The bubbles you see from the reaction come from the carbon dioxide escaping the solution that is left. Carbon dioxide is heavier than air, so, it flows almost like water when it overflows the container. It is a gas that you exhale (though in small amounts), because it is a product of the reactions that keep your body going.

What's left is a dilute solution of sodium acetate in water.

Chemical reactions:

Acetic Acid: CH3COOH -> CH3COO- + H+

Sodium Bicarbonate: NaHCO3 -> Na+ + HCO3 -

H++ HCO3- -> H2CO3 Carbonic Acid

H2CO3 -> H2O + CO2

It will "explode" in a way that most kids with those materials to hand could tell you about.