Quantum gravity is a collective name for the latest developments in gravitation theory, trying to unify general relativity and quantum mechanics. General relativity is coping very well at the macroscopic level and the latter is a theory of the microscopic scales and smaller. The two most promising directions seem to be 'string theory' and 'loop quantum gravity' (LQG for short).
String theory postulates very tiny string-like objects that vibrate in different modes and give rise to the elementary particles and the basic forces of nature. It includes the graviton, which is a virtual particle that can describe the effects of gravity.
One of the problems with string theory is that it assumes a fixed background spacetime, which is somewhat in conflict with relativity theory. Various varieties of string theory are nowadays usually combined under the collective name M-theory.
Loop Quantum Gravity (LQC)
LQC is an effort to formulate background-independent theory. It preserves many of the important features of general relativity, while at the same time employing quantization of both space and time at the Planck scale. Quantization basically means that there is a fundamental 'packet' of something that cannot be broken down further. Planck time and Planck length are the two fundamental packets of time and of space respectively.
There have also been difficulties with LQG, amongst others that it has one crucial free parameter that has to be chosen in order to give results compatible with both general relativity and quantum physics. LQG does however give rise to gravitons naturally and allows them to interact as expected, reproducing Newton's/Einstein's gravity correctly.
Both the string theory and LQG branches of quantum gravity are incomplete as to date. It remains to be seen if one of them will come out on top of the quantum gravity stakes and explain physics (including gravity) completely, all the way from fundamental particles up the Universe as a whole. If one of them does, it may become the 'theory of everything'.
In the meantime, we as engineers have a hard time trying to comprehend it at all, but then, does it matter? It may not matter to most of us, but it will be nice to have a theory that explains one of the most fundamental forces of nature - gravity.