When you see a nuclear bomb land somewhere - in movies, documentary footage, pictures - it makes mushroom clouds. The phenomenon is due to something called the Rayleigh-Taylor instability.

It begins with an explosion that creates a Pyrocumulus Cloud. This ball of burning hot gasses accelerates outwardly in all directions. Because the burning ball of accelerated gasses is hotter, it is less dense than the surrounding air, and thus will rise (in the case of nuclear explosions, extremely rapidly, which forms the mushroom cap).

This entire process is something that describes the Rayleigh-Taylor instability. This instability is well known in physics and, in general, describes the merging between two different substances (mainly liquids and gases) that have different densities and are subjected to acceleration. In the case of an atomic bomb, the acceleration, and the hotter gases creating the differing densities of material, are caused by the explosion.

From this, you might have guessed you don't necessarily need an atomic bomb to create a mushroom cloud. All you need is enough energy delivered rapidly (in this case an explosion) that creates a pocket of differing densities of material (in this case, heated gases).

Read more about the phenomenon here.