As described on Wikipedia's section about the Turkish invasion of Cyprus, the operation which was launched on 20 July 1974 was a military operation by Turkey in response to a Greek military-junta-backed coup in Cyprus.

Though it is now 35 years later, the Turkish Army has not been withdrawn from the island, and a subservient 'state' has been created which the army wishes legitimised internationally. 

This situation has caused many hardships to Cypriots:

• Lots of Cypriots are not allowed to go back to their own homes
• The government of the Republic of Cyprus has no control over the north of the island.
• Male Turkish Cypriots abroad can't stay longer than a month in the north because of the compulsory military service even if they're 50 years old.
• The Turkish army in the north has the right to do anything without asking the government.
• The Police department in the north can't sue any member of the Turkish armed forces, even if someone is killed, because the police department is subservient to the Turkish Army.
• Turkey controls the money and economy of the so-called government of 'Turkish' Cyprus.
• Property in the north which belongs to expelled Greek Cypriots is being sold at low prices. (The ECJ judgment in April 2009 has caused such transactions to become much harder)
• Since the free health services in 'northern' Cyprus are overstretched, TC's use private hospitals in the north or prefer crossing to the south for treatment.
• Numbers of students in Government schools of the north whose parents are of Cypriot origin are outnumbered 3 to 1 by children of the settlers from Turkey.  For this reason, many Turkish Cypriots send their children to private schools or schools in the south.
• Turkish settlers in the north have the ability with their votes to determine the outcome of elections there, as they now outnumber TC votes 2 to 1.

On the other hand, NATO, the United Kingdom and Turkey have benefits from it:

• The U.K and Turkey (and NATO because Turkey is in NATO) have their troops on a strategically important island, as envisaged by the originating 'independence agreements' which made Britain, Greece and Turkey into 'guarantor' powers. (the continuing existence of which, or not,  is a 'red-line' issue during current negotiations for a settlement of the Cyprus problem 'By Cypriots for Cypriots'.
• Turkey seeks 'independence' for the north, the right to a say in the south, and acceptance of the ethnic cleansing it created through invasion, in the form of a bizonal and bicommunal settlement while actually talking about two people and two nations, and pursuing Taiwanisation as a solution.
• Turkey has a client state in the north of Cyprus to influence EU matters during its own accession negotiations for entry to the EU.
• Turkey has breached articles of the Geneva Convention on the 'settlement' of 'occupied territory' by its nationals, it has been condemned on numerous occasions over violation of human rights in the territory in northern Cyprus which is under its continuing control.
• Turkish settlers are allowed to work and live in the north, to vote, and they and their children receive free education and health services.

Note: The views expressed above are held by many Cypriots and is addressed at the government of Turkey and the Turkish Army as an institution, and is not intended to cause offence to any individual or to any ethnic group.

So, don't  take offense.