The geographical term 'Roumeli', A.K.A. "Rumeli" is a bastardization of the Ottoman name 'Rum Millet' which meant the 'Greece' controlled by the Ottomans. 'Rum' in turn was borrowed by the Ottomans and comes from the word 'Roman' and is a word used by the Greeks even today to describe themselves: as "Romaioi" or citizens of the Eastern Roman Empire.
While not an officially designated area on maps, Rumeli exists within the Greek psyche (a Greek word and pronounced: spee-hee). Geographically Roumeli may be thought of as Sterea Ellada and up to Igoumanitsa on the west coast which overlaps part of Epirus.
Readers will recall that for over one thousand years during the 2nd half of the empire, the Eastern Roman Empire was ruled from Constantinople and most of the business of the Empire was conducted in the Greek language.
So when Greeks refer to themselves as being Romaioi they do so with a sense of pride due to identification with ten centuries of rule and sophisticated Greek Culture and not as being a subjugated people.
The concept of being a 'Romaios' is slowly dying out what with Television, The E. U. and perhaps most of all 20 years of Socialism doing its utmost to re-define what it means to be a Greek even to the point of denying Greek orthodoxy and the Greek religion as much as possible. All one has to do is look at the flag of Greece to see the absurdity of this notion because in Greece the church is not separate from state. You aren't a real Greek if you aren't Greek Orthodox ~ sorry Carlo!
"Romaiosini" on the other hand is the concept of things that stem from the latter Roman Empire as opposed to those that stem from classical or Hellenistic Greece.