A classical example for a reaction that must be done under anhydrous conditions is the Grignard reaction. (Equation 1)
In the first step of the reaction, the nucleophilic attack of the Grignard reagent RMgX occurs on an electrophile (in this case a ketone). In this step it is imperative that not even the smallest traces of water be present. The Grignard reagent, while a strong nucleophile, is an even stronger base. In the presence of water it will preferentially act as a base and deprotonate water, resulting in the loss of the nucleophilic Grignard reagent and in the formation of an alkane, an undesired byproduct. (Equation 2)