An antacid is just a weak base. When you add a weak base to a strong acid (or strong base to a weak acid), you get a buffer! The reason why you would want a weak base to fight a stomach ache is because stomach aches are usually caused by the acidic environment in your stomach irritating the lining of your stomach or esophagus. By adding a base, you reduce the acidity and counteract these effects.

Let's break it down even more:
A strong acid/base is defined as a molecule that readily dissociates. In your stomach, for example, the primary acid is HCl (Hydrochloric Acid) , which dissociates into one H+ and one Cl- ion.

A weak acid/base does not dissociate readily. Antacids are usually compounds like sodium bicarbonate - aka baking soda - (NaHCO₃) which dissociates into Na+ and HCO_3-. This is considered weak because the molecule doesn't break all the way down (so that each element is on its own).

All of these reactions take place in an aqueous (watery) environment. Lets see what each dissociation reaction looks like on its own:
1. HCl <--> H+ + Cl-
2. NaHCO₃ <--> Na+ + HCO_3-
In the first reaction (the one that naturally occurs in your stomach), dissociation of HCl in water generates H+ ions, which lowers the acidity and occasionally leads to stomach pain. A negative ion (Cl-) is also produced.
In the second reaction, sodium bicarbonate dissociates into a positive ion (Na+) and its conjugate acid (HCO_3-).
What happens if we combine these two reactions? The answer is kind of magical when you think about it!:
3. HCl + NaHCO₃ <--> H₂CO₃ + NaCl <--> H₂O + CO₂ + NaCl
The free proton from the first reaction is attracted to the negatively charged carbonic acid produced in the second reaction. This creates bicarbonate, which is unstable, and eventually becomes CO₂ (a gas that you can burp out of your system) and water. The free ions, Na+ and Cl-, naturally come together to create salt. In this process, you remove the free protons which lowers the overall acidity of the solution.

One definition of an acid is a substance which increases the concentration of H+ (in practice, actually H₃O+) ions in an aqueous solution. When an acid dissolves in water, it dissociates into a positively charged H+ ion and some negatively-charged anion, with the reaction [rxn1]
HA -> H+ + A- (A- signifying some anion).

(The H+ ion readily combines with a nearby water molecule to form H₃0+, but that is a relatively minor detail here.) Antacids are examples of bases, which are substances that increase the concentration of OH- ions in a solution. Dissolving a base results in a dissociation analogous to that of acids, into some positively-charged cation and the OH- ion: [rxn2]
BOH -> B+ + OH- (B+ representing the cation). (The OH- does not combine with a water molecule though.) The H₃O+ and OH- ions are highly reactive and combine with each other to form H₂O via the reaction [rxn3]
H₃O+ + OH- -> 2*H₂O.

The overall reaction is a combination of the previous three and may be written as
HA + BOH -> H₂O + AB. This is called a neutralization reaction. The anion from the acid and the cation from the antacid may form a compound AB, but in practice the result is an innocuous salt whose ions remains dissociated in the aqueous solution.

(Antacids are designed to neutralize stomach acid, for which the anion is Cl- (stomach is hydrochloric acid, HCl) and the cation from the antacid is some kind of metal, such as Na+ or Ca₂₊. The ionic compounds which would otherwise form, such as NaCl, dissolve in water and so remain in solution.)

It varies, but usually antacids contain sodium bicarbonate and a weak acid such as citric acid. Both of these dissolve in water, and once they do so, the sodium bicarbonate reacts with the citric acid to make carbon dioxide and citrate ion (it's the same reaction as vinegar with baking soda, except using citric acid instead of acetic acid). The citrate ion then takes up the protons from hydrochloric acid, creating chloride ion and citric acid. Citric acid is a weak acid compared to hydrochloric acid, and so doesn't cause as much damage and isn't as painful.

The reaction between an acid and an antacid is actually called an acid-base reaction in chemistry. When an acid and base react together, they neutralize each other and create a salt. For example, if you took hydrogen chloride (an acid) and mixed it with sodium hydroxide (a base), they would eventually create water (neutral) and sodium chloride, which is table salt! Chemically, it has to do with the transferring of hydrogens. Acids want to give away their hydrogens, and bases want to steal hydrogens. So when an acid and base are mixed together, they trade hydrogens and neutralize!