If you have an emergent talent for science, your teacher might bump you up to an advanced science class. Emergent is an adjective that describes something that is emerging, or suddenly coming into existence.
Emergent means “coming into being.” It's often used in phrases like “emergent technologies.” These are brand-new technologies that we can expect to be widely used in the near future. The Internet, for example, was an emergent technology in the early 1990s. Emergent sometimes implies that what is coming into being is surprising and demands a response. An “emergent disease,” for instance, would send scientists scrambling to find a cure. In this sense emergent is related to emergency.