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In science an “Emergent” property is simply put a property that does not exist at smaller scales. An example of an emergent property that is mostly used to explain “emergence” is: temperature.
Temperature is not something that exists on the molecular level. At the molecular level there only exists movement and/or vibration and this movement and/or vibration is what is perceived at larger scales as temperature.
Another example of emergence is the property “solidness”. In Quantum Field Theory, solidness is not something that exists at the “quantum” scale. At the quantum scale there only exist fields that (in this case) repel. This repelling effect then what we perceive at larger scales as solidness.
So what they mean with “emergent gravity” is that what we perceive as gravity does not exist at a more fundamental level. This gravity that we perceive it is as they say “emergent” from something else.
There is one physicist in particular (Erik Verlinde) who is the first to suggest that gravity could be emergent. That is including that what is now called dark matter. His idea is basically that gravity and dark matter are emergent from dark energy.
This emergent gravity idea is an attempt to make a modified gravity law, but this has no relation to other modified gravity laws like MOND like others are suggesting in their answer.
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