A simple, non-anthropologist's explanation is that the close family is the smallest and closest segment, and will generally stand with each other. That family is also a part of a larger segment of more distant cousins and their families, who will stand with each other when attacked by outsiders. They are then part of larger segments with the same characteristics. Basically, brothers will fight against cousins, unless outsiders come, and then they will join together, as expressed eg. in the old Arab saying: "Me against my brothers, me and my brothers against my cousins, me and my cousins against the world."
The ancient Hebrew nation (the Israelites) is one of the more well-known examples, with 12 tribes originating from one common ancestor (Jacob). The largest segmentary lineage society today is believed to be the Pashtun people of Afghanistan and Pakistan, with some 42 million members organized into a vast tribal structure.