My understanding is that the area known as Macedonia is most largely in Greece with smaller parts in Albania, Bulgaria and what was Yugoslavia. In ancient times, when there were no countries, people identified with one another through language and culture. The people in Macedonia region shared what we now call ancient Greek language and culture, like Athenians or Spartans. This included Philip II of Macedon and his son, Alexander the Great.
The people now living in the Former Yugoslav Republic of Macedonia are mostly of Slavic origin and descended from settlers in the 6th and 7th centuries AD. The present 'Macedonian' language derives from Bulgarian / Slavic language.
FYROM was known as Southern Serbia and then Vardar Banovina until after World War II when Tito renamed it Macedonia and expressed the aim to unite with it the rest of ancient Macedonia, i.e. to take control of the parts of ancient Macedonia now in Albania, Bulgaria and Greece.
These claims resurfaced in the 1980s and 90s and official FYROM documents and school books refer to 'Greater Macedonia' including the city of Thessaloniki and Mount Olympus, home of the ancient Greek gods. Additionally, the ancient Greek culture of Macedonia has been claimed by FYROM, e.g. a statue of Alexander the Great was erected in the capital, Skopje, and Skopje airport was renamed Skopje "Alexander the Great" Airport, despite Kavala airport in Greece already being called Kavala International Airport "Megas Alexandros".
FYROM occupies a land area of 25,713 km2, the Greek region of Macedonia occupies a land area of 34,177 km2 .
The population of FYROM is around 2.1 million, of which 25% are of Albanian descent and class themselves as Albanians, not 'Macedonians'. The population of the Macedonia region in Greece is around 2.5 million, including ethnic groups such as Albanians, Roma, Aromanians and Slavic speakers - but relatively small numbers.
More recently the FYROM government has backed down on certain claims to territory and culture but there's no guarantee these claims won't resurface if a nationalist government is elected in the future.