Vinegar contains acetic acid (CH₃COOH) and baking soda contains sodium bicarbonate (NaHCO₃), which is a base. When mixed, they react to form carbon dioxide (CO₂), water (H₂O), and sodium acetate (CH₃COONa). The oxygen atoms in the acetic acid are quite electronegative, and they would like to get rid of the extra H atom to have the electrons all to themselves, which creates a free proton (H+) and an acetate anion (CH₃COO-). If this proton finds a bicarbonate anion (HCO_3-), then it can produce carbonic acid (H₂CO₃).

Carbonic acid is fairly unstable in water because it can split into carbon dioxide (CO₂) and water (H₂O), both of which are very stable and allow the oxygen atoms to have small partial negative charges because of their electronegativity. The carbon dioxide typically escapes from the mixture as a gas, which is what causes the bubbling to occur when you mix baking soda and vinegar.