In 1995 Phil Anderson organised a colloquium: "Physics: the opening to complexity" for the USA National Academy of Sciences. His introduction is worth reading. I reproduce below an extract because it gives such a clear discussion of emergence.
At this frontier [of complexity], the watchword is not reductionism but emergence. Emergent complex phenomena are by no means in violation of the microscopic laws, but they do not appear as logicaly consequent on these laws. That this is the case can be illustrated by two examples which show that a complex phenomenon can follow laws independently of the detailed substrate in which it is expressed.
(i) The "Bardeen-Cooper-Schrieffer (BCS)" phenomenon of broken gauge symmetry in dense Fermion liquids has at least three expressions: electrons in metals, of course, where it is called "superconductivity"; 3He atoms, which become a superfluid when liquid 3He is cooled below 1-3 mK; and nucleons both in ordinary nuclei (the pairing phenomenon explained by Bohr, Mottelson, and Pines) and in neutron stars, on a giant scale, where superfluidity is responsible for the "glitch" phenomenon. All of these different physical embodients obey the general laws of broken symmetry that are the canonical example of emergence in physics.
(ii) One may make a digital computer using electrical relays, vacuum tubes, transistors, or neurons; .... the rules governing computation do not vary depending on the physical substrate in which they are expressed; hence, they are logically independent of the physical laws governing that substrate.The picture is the ENIAC.