Statistical Regression is a technique used to determine how a variable of interest, or a dependent variable, is affected by one or more independent variables. Basically, Statistical Regression answers the question: What will be the value of Y (the dependent variable) if I change the value of X (the independent variable)?
For example, let's say you want to find out whether length of breastfeeding is related to a child's IQ. If you were to do a Statistical Regression, the x-axis would be the length of time a child was breastfed, and the y-axis would represent the child's IQ score. Every participant in the study would be recorded as a dot on a graph, representing the intersection of the X and Y variables. After recording all your data, you will end up with a bunch of dots scattered all over your graph.
What Statistical Regression does is it determines to draw a line that is closest to all of the dots on your graph. This is known as the line of best fit, and it tells you by how much Y changes with every change in X. In this case, given how long a child is breastfed, it enables you to predict what his IQ score would be.