In a broad sense all theaters of war during the First World War saw the use of trenches in some form or another. However, no theater saw the same extensive use of trenches that the Western Front did except maybe the Gallipoli front.

Trenches were not a new feature of war when the First World War broke out. They were used extensively during the Russo-Japanese War a decade earlier, they saw use during the Crimean War American Civil War so the concepot wasn't new. Basically, if hot lead is whizzing about across a battlefield, one of the safest places to be is below ground level, its a basic human instinct to seek safety.

The reason trench wafre became such a prominent aspect of the conflict on the Western Front is due to the high density of troop formations, the high ration of artillery to the length of the front and the relative shortness of the front between the Swiss Border and the English channel which limited the amount of maneuvering the various armies could do before they ran out of land. Similarly, the campaign on Gallipoli saw incredibly difficult terrain that limited maneuvering. On both fronts, as armies came up against opposition and rifles, machine guns and artillery made staying above ground incredibly dangerous, soldiers sought shelter below ground.

The same cannot be said about the Eastern Front which saw a much, much, much larger geographical area for armies to maneuver. Due to the length of this front as well, more men were needed to hold the front line. This usually stretched the manpower requirements of the armies and so the front was thinner then it was on the Western Front. As such, it was more easily penetrated during an attack and was not as stagnant as the front in France and Belgium. This led to more fluidity so trenches were more temporary then in the West.

In the Middle East also, trenches were a feature but not a majorly common one. Again, the geography as well as the use of horses saw a far more mobile type of warfare that limited the usefulness of static defences such as trenches...plus dry sand is notoriously difficult to dig trenches in effectively without a major use of supports.

The Italian Front saw the widespread use of trenchworks though these were accompanied by tunnels dug into the mountainside through the rock and ice. This was a major stalemated front as well but doesn't feature as majorly in the popular history of the war so doesn't get associated with trench warfare like the Western Front does.