Physicists who study quantum gravity are trying to find a description of gravity on the very smallest length scales of the universe; smaller than the size of an atom or electron.
The theory that describes gravity, explains how objects that have mass “curve spacetime”. This is similar to how a ball, placed in the centre of a blanket, will curve the blanket if it is held up at the edges. On the other hand, the theory that describes everything at very small length scales, quantum theory, only works when this blanket is completely flat. It does not tolerate any curvature. And this poses problems, because there is no theory (yet) that describes both correctly at the same time.
It is recommended that you read the sections on General Relativity and Quantum Mechanics first
As explained in the Introduction to General Relativity, the gravitational force is nowadays understood as arising from the curving of spacetime, induced by mass or energy. This is especially relevant at large scales and distances. For effects in everyday life and for smaller distances, it is sufficient to describe gravity as an attractive force between two objects. This is according to the theory as developed by Newton over 300 years ago. This does not mean that General Relativity can’t be used, it will just give exactly the same results as Newton’s theory, as the differences between the two theories become negligible.
Going to even smaller distances, like sub-atomic length scales, we enter the world described by quantum mechanics. This theory only works in a flat space. Curving space gives rise to many mathematical challenges and most of the ideas that quantum mechanics is based on are not applicable any more.
For example: there are an infinite number of parameters that control the strength of the gravitational force when space is curved. This means that there cannot be an experiment to find out what values these parameters should be.
Further, time plays a different role in quantum theory and General Relativity. Time is not an absolute quantity in General Relativity as it changes depending on how much mass and energy is present. However, in quantum mechanics time is a constant “in the background”; it keeps on ticking and quantum systems evolve from one state to the other. There is no interaction between these systems and time that can change how fast or slow the time runs.
All other fundamental forces in nature can be described on quantum scales, it is only for gravity that there is no good description yet. Quantum Gravity is the field of physics that tries to find exactly this: a description for gravity that complies with quantum mechanics. (Note: this is not the same as the effort to unify all forces into one mathematical description!)