How to understand locality and non-locality in Quantum Mechanics? What actually is the definition of locality and non-locality?
Does non-locality in Quantum Mechanics mean however far you separate 2 entangled atoms in space, the 2 atoms can still influence each other? Doesn't non-locality mean action at distance? If not, what's the difference between them?
 A: The short answer is yes, in quantum mechanics quantum non-locality refers to the apparent instantaneous propagation of correlations between entangled systems, irrespective of their spatial separation. In quantum field theory, the notion of locality may have a different meaning, as pointed out already in a comment. 
Details:
The notions of locality and non-locality in Quantum Mechanics have been originally defined in the context of the EPR controversy between Einstein and Bohr on the phenomenon of quantum entanglement. 
Basically, the general "principle of locality" (Wikipedia ref.) requires that "for an action at one point to have an influence at another point, something in the space between the points, such as a field, must mediate the action". In view of the theory of relativity, the speed at which such an action, interaction, or influence can be transmitted between distant points in space cannot exceed the speed of light. This formulation is also known as "Einstein locality" or "local relativistic causality". It is often stated as "nothing can propagate faster than light, be it energy or merely information" or simply "no spooky action-at-a-distance", as Einstein himself put it. For the past 20 years or so it has been referred to also as the "no-signaling" condition.
The phenomenon of entanglement between quantum systems raised the non-locality problem first noted in the EPR paper: A projective measurement on a quantum system at one space location instantly collapses the state of an entangled counterpart at a distant location. Quantum mechanical non-locality refers to this apparent entanglement-mediated violation of Einstein locality. 
The remarkable thing about quantum non-locality, however, is that it actually does not imply violation of relativistic causality. Although entanglement correlations are affected instantaneously, they cannot be harnessed for faster-than-light communications. The reason is that the outcome of the local projective measurement is itself statistic and cannot be predicted beforehand. If the same kind of measurement is performed on multiple copies of identically prepared pairs of entangled systems, the overall statistical result is that locally both systems conform to the statistics prescribed by their respective local quantum states. The "spooky action-at-a-distance" of the distant measurement gets wiped out in the total statistics.
There is already a host of related questions on Physics.SE. See for instance Quantum entanglement and spooky action at a distance and similar.    
