Prism compares slopes of two or more regression lines if you check the option: "Test whether the slopes and intercepts are significantly different". If selected, a summary of this analysis will be presented on the results tab titled "Are lines different?"
Prism compares slopes first. It calculates a P value (two-tailed) testing the null hypothesis that the slopes are all identical (the lines are parallel). The P value answers this question:
If the slopes really were identical, what is the chance that randomly selected data points would have slopes as different (or more different) than you observed.
If the P value is low, Prism concludes that the lines are significantly different. In that case, there is no point in comparing the intercepts. The intersection point of two lines is:
If the P value is high, Prism concludes that the slopes are not significantly different and calculates a single slope for all the lines. Essentially, it shares the Slope parameter between the two data sets.
If the slopes are significantly different, there is no point comparing intercepts. If the slopes are indistinguishable, the lines could be parallel with distinct intercepts. Or the lines could be identical. with the same slopes and intercepts.
Prism calculates a second P value testing the null hypothesis that the lines are identical. If this P value is low, conclude that the lines are not identical (they are distinct but parallel). If this second P value is high, there is no compelling evidence that the lines are different. It does this by sharing the slopes (so they are forced to be the same) and then it compares elevations. With equal slopes, comparing elevations tests if the lines are identical. It doesn't matter if you compare the elevation at X=0 or X=any other value.
This method is equivalent to an Analysis of Covariance (ANCOVA), although ANCOVA can be extended to more complicated situations. It also is equivalent to using Prism's nonlinear regression analysis with a straight-line model, and using an F test to compare a global model where slope is shared among the data sets with a model where each dataset gets its own slope.
Chapter 18 of J Zar, Biostatistical Analysis, 2nd edition, Prentice-Hall, 1984.