Poincaré's conjecture is an assertion about certain three-dimensional shapes. We say a shape is simply connected if any loop drawn in the shape can be pulled closed to a point without leaving the shape. The surface of a sphere, for instance, is simply connected, but the surface of a doughnut is not; a loop along the outer "equator" of the doughnut can't be contracted to a point unless it departs, at some moment, from the doughnut's surface. The property of being simply connected, the reader will note, doesn't depend on how big or small the shape is; and it doesn't change if the shape is bent, twisted, or otherwise deformed. You might say the property is quite robust; and the study of such robust properties of shapes is the mathematical field of topology, which Poincaré more or less invented in the late 19th century. (The reader who's seen other nontechnical accounts of the subject will forgive me, I hope, for perpetuating the fiction that the whole field of topology is actually confined to the study of spheres and doughnuts. There are other shapes, I promise: They're just harder to describe.)
Poincaré was able to prove that if a two-dimensional surface was simply connected, it was automatically some bent, twisted, and deformed version of the sphere. His conjecture—phrased so loosely that I don't advise you to think about it yourself without further reading—is that the same is true for three-dimensional shapes. Poincaré couldn't have made this conjecture absent his years of study of topology, or the earlier theorems he'd carefully proved, or the earlier conjectures on the same theme he'd tried out and found to be false. But neither could he have made such a bold guess without the kind of wild intuition he valued above all. It's only in the presence of both conditions—deduction and inspiration, long experience and youthful audacity—that new math gets made, as it was made by Perelman, and as it was made on the day Poincaré wrote down his conjecture. He was 50 years old.