In physics, emergent properties are properties that exist on a macroscopic scale but not on the microscopic scale. It can also be described as a property or phenomena that cannot be predicted from the properties of its consituents. It resembles the saying that "the whole is more than the sum of its parts" by describing complex interactions and properties of collectives. A popular human example would be the mind. We cannot find traces of consciousness in the synapses of the brain, and yet when the structure of the brain exists, consciousness exists. Another example would be a cake. When you look at the ingredients to a cake, you cannot find the properties of "cake", and yet, when the ingredients mix and interact and react (ie. you mix and bake them), the properties of "cake" emerge.
When complex results arise from a combination of simple causes. An idea much liked by chaos theorists, who like to build simple computer models and imply that reality is equally predictable. Though this may work for (e.g) predicting the shape of ants' nests, it doesn't work for human society, partly because of reflexivity.