So many candles are made from “paraffin wax” which has the chemical formula C₂₅H₅₂. When paraffin comes into contact with oxygen with enough energy, it burns completely and becomes water and carbon dioxide. The fire is a result of this reaction happening very quickly and is known as combustion. The fire is so hot that the wax actually turns from a solid to a vapor. It is the wax vapor that is actually participating in the chemical reaction.

A candle is made of wax, which is made of hydrocarbons. This means that it's composed of compounds with a carbon backbone and hydrogen atoms that dangle off of the backbone. Other examples of hydrocarbons are gasoline, propane, and even things like paper and plant matter are largely composed of hydrocarbon type compounds. The chemical reaction involved in burning is the oxidation of these hydrocarbons. In an ideal reaction, the reactants are the hydrocarbon and oxygen and the products are water, carbon dioxide and energy (in the form of light and heat).

In reality, burning is not totally ideal and there are additionally side reactions and secondary products like nitrogen oxides (from reactions with air) and partially oxidized hydrocarbons (like ash).

The candle is made of wax, which is a combination of different kinds of hydrocarbons (i.e. CxHy, where y = 2x + 2). x and y are large enough that the mixture is solid at room temperature; I don't know what numbers this requires, but I'd guess x would be at least 12 or so (since x=8, a.k.a. octane, is liquid at room temperature).

The reaction combines atmospheric oxygen with the hydrocarbons to create carbon dioxide and water just the same way as burning any other hydrocarbon (e.g. for methane, 2O₂ + CH₄ -> CO₂ + 2H₂O).