Trying to be very concise here:
- Quantum Mechanics brings minimal uncertainty (fluctuations, fuzziness) to quantities.
- According to general relativity, gravity is an effect of mass curving spacetime.
- Using quantum mechanics or quantum field theories (such as QED, the quantized theory of electromagnetism) in any fixed geometry (Euclidean/flat as in the non-relativistic limit, but also in a fixed curvature) is straightforward.
- It becomes troublesome when masses with an uncertain position curve spacetime: this means that the curvature of spacetime is itself uncertain! This makes it hard to even work in a fixed coordinate system, and gives complicated backactions between the spacetime and the massive particles.
Renormalisation is a very technical issue of making sense of infinities in all quantum field theories, but the above is the main conceptual difference that distinguishes gravity from the other fundamental interactions (electromagnetism and the strong and weak interaction), that are described by fluctuating fields in a fixed geometry.