Armenians who are considered Eastern Armenian, and who speak Eastern Armenian dialects are basically those who originate from the former Russian and Persian Empires (basically, the the former USSR and Iran).
Iranian Armenians still write in the original spelling system invented by Mesrop Mashtots, but in the former USSR a new, simplified spelling system is used which was introduced in Soviet times.
Much of the modern Eastern Armenian dialect originates from Tbilisi, which was the center of Armenian culture in the Russian Empire. With the establishment of the independent Armenian state in 1918, and then the Soviet Armenian state in 1921, the center of Armenian culture in the Caucasus rapidly moved to Yerevan. Numerous Eastern Armenian dialects are still spoken today across Armenia, Karabakh, Georgia and Russia aside from the standardized Yerevanian Armenian.
See also Western Armenian.