Why the desire to combine relativity theory (i.e. gravity) with quantum theory?
First, such a unified theory will be needed to answer many profound questions, including: How did the Big Bang begin? Or: What is happening deep inside a black hole? But more importantly, the history of physics has taught us again and again that successful unifications of seemingly disparate theories are a natural direction of progress in physics, invariably leading to deeper insights into—and more profound questions about—the workings of our mysterious universe. The search for a quantum theory of gravity, which strongly challenges the entire foundation of our understanding of the universe, promises to be the greatest unification of all. If you want another non-technical explanation, read this.