Refers to a mathematical generalisation of the Galton-Watson model. From T.E. Harris, The Theory of Branching Processes ©1963 Springer-Verlag:
"We may think of a branching process as a mathematical representation of the development of a population whose members reproduce and die, subject to the laws of chance. The objects may be of different types, depending on their age, energy, position, or other factors. However, they must not interfere with one another. This assumption, which unifies the mathematical theory, seems justified for some populations of physical particles such as neutrons or cosmic rays, but only under very restricted circumstances for biological populations."
Suppose that there is a population of bunnies on the Meadow of Infinite Abundance. Each bunny has some probablity of having no offspring, some other of having one, and so on. The Meadow is so big and lush with bunny chow that these probabilities don't change no matter how big the population gets (they might even be asexual bunnies). The nice thing is that the dynamics of an individual is sufficient for studying the dynamics of the whole bunny population. One can then easily obtain the probability of extinction (no more bunnies) or some asymptotic distribution of sizes for the population.
Branching processes have been very useful in the discipline of population genetics for studying the frequency dynamics of alleles. See Muller's ratchet.