Rum < Roman is the traditional Ottoman designation for Orthodox Christians in the Ottoman Empire, inherited from the self-description of the Byzantine Empire, and it continues to be the Turkish designation for ethnic Greeks, living in Turkey and Cyprus.
Urum is a variant of Rum, and is used as the self-designation of several Turkic speakers who are traditionally Greek Orthodox, and who are believed to have been linguistically assimilated. Notable among them are the Urum who live in Mariupol’, in the Ukraine, and who moved there along with Greek-speaking Christians from the Crimea in 1778.
Yunan is the Turkish term derived from the Persian word for Greeks, which in turn derives from Ionian. In Ottoman times, it was used to refer to Ancient Greeks, just as Hellene was in Greek. After the independence of the Greek state, Yunan also came to be used for Greeks in the new state (just as Hellene was in Greece), while Rum continued to be used for Greeks in the Ottoman Empire (unlike Greece, which had an irredentist vision). The country of the Yunan in Turkey, Greece, is Yunanistan.