Dictionary Definition :

“An molecule or group having a tendency to attract or acquire electrons.”

Selected from OxfordDictionaries.com

Functional Definition: 

Electrophiles are electron pair acceptors. Electrophiles are usually positively charged or have a partial positive charge, and they can also be double bonds.

While the reaction mechanism bellow may seem complicated it is demonstrating  how a double bond can break and act as an electrophile. The red arrow shows where the bond is broken and to which carbon the A attaches. The rest of the mechanism shows the formation of new bonds and the movement of charge.

Figure courtesy of Manus Manske of Wikipedia Commons