This famous Persian poet and mystic was named Jalāl ad-Dīn Muḥammad Balkhī (جلالالدین محمد بلخى) or Jalāl ad-Dīn Muḥammad Rūmī (جلالالدین محمد رومی), where Jalāl ad-Dīn Muḥammad was his actual personal name while Balkhī and Rūmī were nisbas referring respectively to his birthplace and the place where he spent much of his later life.
In the English-speaking world, he is usually referred to as Rumi after one of these nisbas. In various Middle Eastern countries (e.g. Turkey), he is commonly referred to as Mevlana or some other variant of Mawlānā (مولانا) meaning "our master".
Why have these sobriquets in particular become popular for him? Nobody refers to him by his personal name; instead we have, in the west, the name "Rumi" meaning "from the western part [of the Persian Empire] which was formerly in the Roman Empire", and, in the Middle East, some variant of the name "Mawlana" meaning "my/our master". How did these names become the standard way of referring to him? And why different names in different countries?