In Madison's definition, a number of citizens, whether a majority or a minority, united by a common impulse of passion or interest adverse to the rights of other citizens or to the community's permanent interests.
In more detail
In Federalist No. 10 Madison argued that the causes of faction cannot be removed without destroying liberty, so its effects must be controlled. He held that a pure democracy offers no cure for the mischiefs of faction and favoured a representative republic over a small direct democracy.