Generating a slope field is a simple way to think about the solution to a first order differential equation.  Consider the differential equation

$ {{dy \over dx} = f(x,y)} $.

Given a point (x,y), the function f gives the slope, m, of the differential equation. If you do this for a set of points , and draw a short line at each point to represent the slope, you can generate a slope field. By choosing a starting point and constructing a line whose slope is m at each point, you can construct a solution curve.