A fractal can be described as a pattern that contains itself on smaller and larger scales. One simple example of a fractal is a tree; it contains the same pattern in the trunk, branches, and leaves. Each time the tree branches it forms a new iteration of its fractal shape.
Systems form patterns which act as a components on a higher scale. When the type of interaction between components is the same on multiple scales, a fractal emerges.
Biological systems appear to have a fractal property, although it is more complex than a simple geometry. The defining pattern of the fractal of biology is specialization and cooperation. At each scale, organisms are composed of a variety of specialized components that behave as machines; they accomplish very specific tasks. These components cooperate to create conditions that allow for the replication of themselves and each other.
The smallest scale at which this pattern occurs is in the biochemical molecules of a prokaryotic cell. The specialized machines are the DNA, RNA, ribozymes, ribosomes, and proteins. Each biomolecule has a specific task, DNA stores information, RNA communicates it, ribosomes translate it into proteins. The proteins do many more tasks, they replicate DNA, recycle other proteins, harvest energy, and protect the cell. The interaction of all of these specialized components generates behavior to maintain homeostasis. The molecules have synergy; they create at dynamic pattern that could not be predicted from any of them individually. The whole is greater than the sum of its parts.
The next scale at which this pattern occurs is within the Eukaryotic cell. Here the specialized components are the organelles; the nucleus, mitochondria, endoplasmic reticulum, golgi complex, centrioles, lysosomes etc. Each organelle accomplishes specific tasks that contribute to the wellbeing of the cell. The nucleus coordinates behavior, the mitochondria provide energy, the ER synthesizes proteins, the golgi sorts them, the centrioles create movement and structure, and lysozomes recycle organelles.
The pattern occurs again on the scale of tissues. Within each tissue there are a variety of specialized cell types that cooperate to survive.
Going up further, there are a variety of specialized tissues within each organ, and specialized organs that compose each organism.
This is the scale we exist on, but it is not the end of the fractal of biology. Within society, humans and some other species specialize at different tasks. Human specialization and cooperation allows for greater efficiency and survival. This results in the meta-organism of a community, town or city. The fractal goes further up to include specialized entities such as corporations, cities, nations, ecosystems and biomes.
The largest scale of the fractal of life is the biosphere of the planet as a whole. Earth is essentially one giant fractal organism.