Fallout, technically, is the effect of radioactive particles (usually from a nuclear detonation, but
not always) 'falling out' of the atmosphere. This effect of nuclear events is usually the farthest-reaching, and in the case of low-order events like
reactor leaks, the most dangerous and destructive.
The type of particles that are distributed will vary depending on the cause. In the case of reactor releases, the radioactive material is typically steam (unless the leak is compounded by other disasters such as the fire at Chernobyl). The steam, which is composed of radioactive water and possibly other contaminants, will eventually cool. Droplets of it will mix with atmospheric humidity, or just condense themselves, and fall as rain or mist. This is the fallout.
In the case of a nuclear or thermonuclear detonation, the fallout is usually composed of materials sucked up into the mushroom cloud and irradiated falling back to earth. Mostly this will be dust which results from thousands of tons of earth and debris being lofted and irradiated. There is typically a negative correlation between the area of damage a nuclear explosion is intended to cause and the amount of fallout it produces. This counterintuitive relationship is due to the fact that in order to do widespread damage with a nuclear weapon, targeters will use airbursts to spread lower (but still massive) blast and thermal effects over a wide area. With an airburst, however, the center of the explosion is several thousand feet above the ground, which is the nearest source of debris and soil to irradiate and distribute. As a result, most of the fallout from an airburst is the remains of the weapon itself, which is tiny compared to the tons of material tossed about when the weapon is set off near ground level in a groundburst.