An array factor is a mathematical factor in the directivity equation of an array antenna. It is the factor by which the directivity function of an individual antenna must be multiplied to get the directivity of the entire array.
An array antenna is a set of multiple identical antennas which work together as a single antenna. The individual antenna elements are connected to a single receiver or transmitter by feedlines that feed the power to the elements in a specific phase relationship. The radio waves from the individual antennas combine and superpose, enhancing radiation in certain directions while suppressing radiation in other directions. The purpose of using multiple antennas in this way is to produce a radiation pattern that could not be achieved by a single antenna. Usually the goal is higher gain (directivity), more of the power radiated in a specific direction.
The radiation pattern of an individual antenna element in isolation is given by a function