To the left the red line with the blue dots is a Bezier curve. Actually this curve is made up of several short straight line segments that connect the blue dots.
The white dots are said to be the control points. The yellow lines connect the control points.
You can drag the control points with your mouse so that you can see how moving them changes the shape of the Bezier curve. Try this, and then read more below.
A Bezier curve is a curve generated under the control of other points. In this example there are four control points, colored white. In a very, very broad sense, the Bezier curve an approximation to a curve that passes through the four control points. There are different methods for forming such an approximate curve. The one used here is called a Bezier-Bernstein approximation.
One can construct a smooth curve from a few data points. These data points are often called control points. The Bezier-Bernstein method used above gives an approximation to a mathematical function which passes through the control points. Such approaches are useful in fields where curves must be designed to meet aesthetic goals. Approximations like this, however, would be of limited value if one needed an accurate description of a function which really contained the data points.
The red curve above is actually made from twenty smaller straight lines that have been joined together at the blue dots. The positions of the blue dots are determined by polynomial functions of the third degree (or cubic functions). The functions used to generate the curve above are Bernstein polynomials of degree three.