An automation curve is a series of lines connected by control points that typically defines a continuous line. As the curve is traversed from left to right, the line defines the level of the parameter controlled by the automation lane.
There are two types of automation curves: Linear and Discrete. The most common type is Linear, in which the space between any two contiguous control points is continuously interpolated; in other words, the values between any two contiguous control points at any given time is given by the straight line connecting them. The second type of automation curve is Discrete, in which no interpolation between control points is done; whatever value the control point is set at is the value it will yield until it reaches the next control point, at which point it will give that value until the next control point, and so on until there are no more control points.
The curve by itself does nothing; it will only control playback if the lane it resides in is in Play mode.