When performing regression, one might wish to model the effect of an independent variable on the response variable as dependent upon another independent variable. In other words, an interaction term is two or more independent variables strapped together sharing the same coefficient.
A ‘normal’ regression equation might look like: \(y = a + bx_1 + cx_2 + dx_3\)
While a regression equation with an interaction term might look like: \(y = a + bx_1 + cx_2x_3\)
Thus, a change of one element in an interaction affects how the others impact the response variable. Pretty useful when you want the effect of one variable to depend on another.
For example, say you were modeling outdoor temperature in response to time of day and cloud cover. The effect cloud cover has on temperature changes depending on the time of day: with sunlight (day-time) clouds will reflect light back into space, but without sunlight (night-time) they provide an insulating effect. One could model this with an interaction term between cloud cover and time of day.
\(Temp = intercept + constant*(CloudCover*TimeOfDay)\)