Dictionary Definition:

“A molecule or group having a tendency to donate electrons or react at electron-poor sites such as protons.”

Selected from OxfordDictionaries.com

Functional Definition: 

A nucleophile is an electron pair donor. Nucleophiles commonly have negative or partially negative charges.

Nucleophilic reactions are some of the key reactions in chemistry. In the figure bellow H2O is acting as the nucleophile because the oxygen is partially negative and it is attracted to the positively charged carbon.  The oxygen attacks  the primary carbon and forms the end product.

Figure courtesy of Manus Manske of Wikipedia Commons