What is quantum gravity?
"Quantum Gravity" is a current research field in theoretical physics that seeks to resolve apparent contradictions between General Relativity and Quantum Mechanics
"Quantum Gravity" is not yet an established part of the body of physics knowledge - rather it describes currently proposed theories that seek to consistently describe the phenomenon of gravity in terms of quantum mechanics.
Why are such theories sought? Well, in order to describe the four fundamental forces (gravity plus electromagnetism and the strong and weak nuclear interactions) in the same mathematical framework - the other three have proven 'unifiable', and so it feels intuitive that gravity should be too.
At present, there are apparent contradictions between our modern understanding of gravity - which comes from General Relativity - and our modern understanding of quantum effects. These theories seek to resolve these problems.
There's more detail on this on the Wikipedia page on it: