These are answers I've given in the past.
In short, the force-to-space ratio was nowhere near as great in the East as it was in the West. Whereas in 1914 over a million Germans on one side, and over a million French, Belgians and British on the other side were crammed in between the channel and the Swiss Frontier, with numerous terrain multipliers allowing the Germans to economize their forces, in the East the distances were simply too great. Both sides dug trenches and dugouts for their forces in the field, but as in the second World War, these did not take on the same permanence as in the west. Where a mass of forces could be attained (the Russians in Galicia, 1914; the Germans and Austro-Hungarians at Gorlitz-Tarnow, 1915; the Russians in Galicia again in 1916), a breakthrough followed by fairly mobile fighting was usually attained.
There's also some good previous answers on trench warfare that will help you understand why it happened on the Western Front:
The tl;dr is that trench warfare wasn't a tactic that the generals initially wanted to adopt, but it was one that they found themselves using after some stalled offenses (i.e. falling back to good defensive ground to avoid giving up too much of their advances). Due to a combination of the density of the front (so trenches spanned the entirety of the front and couldn't be circumvented) and the defensive effectiveness of the trenches (making it difficult to break through), trench warfare ended up lasting for quite awhile.
On the other hand, as mentioned by DuxBelisarius, the Eastern Front was large enough that there simply wasn't the manpower to have a front made up of an unbroken line of trenches. So trench warfare really couldn't develop.