Physics has been spectacularly successful in creating a mathematical model for a lot of phenomenon in the world like motion of celestial bodies, behaviour of fundamental particles, light, sound and many others. It is possible to use the mathematical equations and predict the behaviour of the real world quite accurately. However, such systems typically have a small number of interacting elements. They can be classified as Simple systems. That does imply that there is anything simple about them. Its just a name to contrast them against a different kind of systems.
An approach that has worked well for understanding such simple systems is Reductionism. Reductionism means that we can understand something by first understanding its parts and their interactions.
However, there are systems where reductionism does not work. An ant colony’s behaviour cannot be understood by understanding the individual ants, a city manifests characteristics that are not intended by any individual, a brain and nervous system consists of simple neurons but collectively exhibit behaviour that are more than sum of its parts. These systems are called Complex systems. A lot of them exhibit some collective behaviour called Emergence.
The science of Complexity aims to uncover common principles that underpin all of these complex systems. Its grand aim is to come up with mathematical equations, much like the case with simple systems, which can be used to understand, predict and may be build a complex system.
Here are two books which serve as excellent introductions to the science of Complexity. My reviews are linked.
Emergence: The Connected Lives of Ants, Brains, Cities, and Software by Steven Johnson
Complexity: A Guided Tour by Melanie Mitchell