Separate regression lines
The problem with fitting separate regression lines by least squares in the different groups is that it is difficult to concisely explain the difference between the groups — the difference between the predicted response in the groups depends on the value of the explanatory variable.
Parallel regression lines
Interpretation is considerably simplified if we constrain the regression lines for the different groups to have the same slope. In the diagram below, the difference between the groups is the same for all values of X.
Parallel lines are not appropriate descriptions of all data sets. Always check a scatterplot first.
Least squares
The principle behind fitting parallel lines to two or more groups is the same as in ordinary simple regression — we choose the parameters to minimise the sum of squared residuals (vertical distances between the data crosses and their corresponding line). The resulting formulae are complicated, but most statistical software will do the calculations for you.