The synthesis of aspirin is actually quite simple. The easiest way (that I know of) is to combine salicylic acid with acetic acid, with an acid catalyst (such as sulfuric acid.) Salicylic acid is a common organic reagent and has an alcohol group on it which will react with the carboxylic acid group of the acetic acid. The reaction produces aspirin (acetylsalicylic acid) along with water as a side product. However, because of the water side product, this method does not work very well. If acetic anhydride is substituted for the acetic acid, the reaction will work much better because the side product formed (acetic acid) is less reactive than water. This type of reaction between an alcohol and a carboxylic acid is called an esterification.

Salicylic acid + acetic anhydride ---> aspirin + acetic acid