Full glossary

Volcanic ash is the term for all fine-grained volcanic products (smaller than 2 mm), normally magma or older rock fragmented during explosive eruptions.

Volcanic ash has nothing to do with fire, but is a mere definition of grain-size. Ash can range in size from sandy to extremely fine; any fragment ejected by a volcano less than 2 mm in diameter is called ash. It may consists of freshly ejected lava (usually turned into a glass shard because of rapid cooling), older fragmented rock, or small crystals.
Ash is produced by explosive activity when expanding gas fragments other material (uprising lava, surrounding rock). The more explosive an eruption, the more ash is usually produced. Hot ash can easily mix with air and erupted gasses and then form an eruption column. If the eruption column is buoyant it will rise to tens of km into the atmosphere during violent eruptions. These ash clouds can then be carried hundreds and thousands of km by wind, even circle the entire hemisphere for a few years before the finest particles are washed out. Ash-loaded eruption columns can also become too dense to rise vertically; instead, they will then collapse to form (usually hot) avalanches, so-called pyroclastic flows.
It is very common to observe that ash particles stick together to form small aggregates, so-called accretionary lapilli, enabling the ash to deposit because of the dramatically increased fall velocity of the aggregate.
Volcanic ash is a serious hazard to life and property; it can cause breathing problems, heavy ash loads on buildings cause the roofs to collapse. Ash plumes in the atmosphere threaten air traffic seriously, because ash even in small concentrations can disable jet engines.