Physicalgeography.net and the Colorado State basic hydraulic page have a good explanation of Horton's Laws.
The first thing that needs to be done in analysing a drainage basin is to classify the stream segments in a basin according to Strahler's method. For each stream order tabulate:
- the number of segments
- the average length of the segments
- the average area of drainage for the segments
According to Physicalgeography.net the bifurcation ratio is the ratio between the number of stream segments in one order and the number of stream segments in the next order.
Horton saw that the bifurcation ratio was always around 3, this he called the Law of Stream Numbers. Apparently a bifurcation ratio of 3 also applies to plant roots, branching of woody plants, veining in leaves and the human circulatory system. I get the impression that without knowing about it, Horton was laying some ground work for fractal mathematics that came to prominence in the 1980s, with the work of Benoit Mandelbrot.
With the Law of Stream Lengths, the average length of stream for each order is calculated, then cumulate lengths found before length ratios between segment orders can be calculated.
Colorado State basic hydraulic page has a more mathematical approach to explaining Horton's Laws, but it's very good. It gives a worked example of how to determine the all the coefficients.
Additional information is also on this site as well.