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
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.
Say you have a scatterplot and you are told to draw a regression line through it. In primary school science class you just roughly draw a line through all the dots. But the problem with this is that every student gets different lines.
Least squares is objective. It minimizes the distance between the dot and the line. The reason why it is squared is because sometimes the dot is below the fitted line and sometimes it is above, which gives negative and positive differences respectively. Squaring it makes it all positive.