Locality in Physics
The world looks simpler when we confine ourselves to local interactions. We affect the world mostly by local interactions. If we want to move something, we usually need to push on it. When Newton discovered his law of universal gravitation, with its action at a distance, that conception of locality was profoundly challenged. He didn't like it, but he could discover no satisfactory hypothesis to explain it. Electricity and magnetism turned out to present similar challenges.
The invention of the electromagnetic field by Faraday and Maxwell changed all that. Field strengths, and the forces they generated were now determined by the fields and charges in the local neighborhood, in effect pervading space with an ether that transmitted the forces. Einstein showed that the ether had to be Lorentz invariant and that gravity too could be localized, with the gravitational field now being determined by the matter and fields in the neighborhood.
One reason this is interesting today is the discovery of the fact that the quantum physics of certain many body systems is radically simplified if the Hamiltonians of those systems have the locality property. These ideas are embodied in so-called tensor networks. In particular the quantum entanglement entropy of such systems can be shown to be proportional to the area of the system boundary. Equally fascinating, effective geometry appears to appear naturally from such tensor networks.
For students of General Relativity and String Theory, this should loudly clang several bells with names like Bekenstein entropy, the Holographic Principle, Maldacena duality, quantum information and spacetime, and Wheeler's 'It from Bit'. These ideas are being actively pursued, and Jennifer Ouellette has a popular level article here in Quanta. A semi-technical article on the fundamental tool, the tensor network is: arxiv.org/abs/1306.2164