The least squares method is the method of drawing the line of best fit through the points on a scatter plot, to show the relationship between two variables. The term "least squares" means that the squared differences between:

-the actual points, and

-the line drawn through them

are minimized.

The regression equation is the equation of this line - i.e., what will the value of the dependent variable (y) be if plug a certain value into the independent variable (x).

Here's a

link to an image of a regression line of best fit drawn through the points in a scatter plot. As you can see, the distances (squared) between the points and the drawn line are kept to a minimum - any other line drawn through the points would not meet this "least squares" criteria.