One interesting definition from Peggy Holman: Emergence disrupts, creates dissonance. We make sense of the disturbances that emergence creates partially through developing language that helps us to tease out useful distinctions. As the vocabulary to describe what is emerging becomes more familiar, our understanding increases. For example, disturbance, disruption, and dissonance are part of the language of engaging emergence. These terms are cousins, and I often use them interchangeably. Disruption is the most general of the three words. If something involves an emotional nuance, chances are that I call the disruption a disturbance. When conflict is involved or the disruption is particularly grating, with a lack of agreement or harmony, I will likely refer to its dissonance.
Or another one from Yaneer Bar-Yam: Emergence refers to the existence or formation of collective behaviors — what parts of a system do together that they would not do alone.
More reading about emergence from Peggy Holman here or from Yaneer Bar-Yam here.