Edit: I don't know who gilded me for this, but thanks! The fact that a month later someone loved it was thanks enough, though :)

Obviously, I have to describe some of the history, and demographics at the time, before we can get into justifications!

Cyprus - Pre-20th Century

This is not going to be an extensively long portion, but it is worth mentioning some of the history here. Cyprus fell into the hands of the Ottomans in the late 16th Century. The Ottoman Empire would later become Turkey, for lack of a better way of describing it. However, the island didn't just become "Ottoman" and lose nationalism of other ethnic groups there. Greek nationalism can be traced as far back as the 18th century. One prominent issue to note is that of Archbishop Kyprianos in 1821 (which is the year when the Greek war of independence from the Ottomans began). He promoted ideas of unifying the island with Greece, and began to spread ideas of Greek nationalism amongst the Greek Cypriots, and was eventually hanged by June of 1821 (along with others who supported him) by the Ottomans. This formed one of the cruxes of the conflict between the Greeks and Turks. The Greeks never really gave up the idea of unifying with Greece's mainland, which would be newly independent after the war, and that persisted.

In 1878, the island's authority was handed over to the British, who also rejected Greek unification proposals. The British were given control in return for assurances that the British would protect the Ottomans against any Russian aggression. Greek Cypriots still hoped for eventual unification with Greece.

Cyprus - The 1950's

There wasn't much going on in the first half of the 20th century; the world wars mostly consumed what went on. However, during this time the two communities (Turks and Greeks) that formed the majority of Cyprus faced numerous problems economic and political (ie. the Great Depression and WWII). But the issue of Cyprus and Greece hit front-and-center again in 1954.

In 1954, Greece's UN representative requested that self-determination be applied to Cyprus, and that it be allowed to join with Greece under said principle. The British rejected this notion and kept it from the general assembly, claiming that Cyprus was the internal problem of the British. This fast-escalated, with the British threatening punitive measures in late 1954 against Greek Cypriots who tried to push for the idea of Greek unification, including prison sentences. In December of 1955, the UNGA decided not to discuss the question of Cyprus, believing it "inappropriate".

However, the UN wasn't the only way the Greek Cypriots were seeking to leave British rule. In April of 1955, EOKA (National Organization of Cypriot Fighters) was formed by Greek nationalists/ex-soldiers. A violent campaign was waged against the British, in some ways similar to what happened in Israel in the years leading up to 1948.

However, during these years, the Turkish Cypriots steadfastly rejected the idea of Greek unification. They said that if the British left, Cyprus should be given over to Turkey instead. But when the EOKA began running violent operations, the Turks formed "Volkan" (an underground organization of their own), which would later morph into the Turkish Resistance Organization (TMT). So now you had the issue of the Greeks, and Turks, wanting to get Cyprus back under control of their respective mainlands. And so they fought both the British, and the other faction, for control. By 1958, this tension was already very apparent. Greece and Turkey themselves were facing rising tensions, and they cooperated and organized talks between themselves on the question of Cyprus' independence. In February of 1959, the two governments agreed to support independence for Cyprus.

Cyprus - The 1960's

In 1960, due to the aforementioned pressures (and pressures from others), the Republic of Cyprus was formed.

However, the Greek Cypriots weren't as happy with this as the Turks. The Turks, for their part, gave up hopes of joining up with Turkey, and their demands for partition of the island, while the Greeks still hoped for Greek unification (President Makarios made this clear in his speech on January 5, 1962). However, the British were now fine with what they'd achieved; they kept bases on Cyprus, which meant they still had the strategic advantage they wanted, but felt they had managed to extricate themselves from the headache of the tensions. During 1961 and 1962, tensions began to rise again due to Greek/Turk tension on the question of Greek unification, leading to the revival of both the EOKA and the TMT. Both started smuggling in weapons and training once more, preparing for conflict.

Sometime in 1963, the Greeks formed a plan to handle these impending hostilities, called the Akritas Plan. The government, which had thus far been pretty divided and inefficient, would have to accept Greek proposals for reform. If the Turks refused, then the Akritas Plan called for their "subjugation" within a day or two; well before any foreign powers could intervene. On November 30th, 1963, President Makarios proposed changes as according to the Akritas Plan that were designed to "remedy the inefficiencies" of the Cyprus government. The 13 points presented included removal of the veto power of the President and VP, which immediately prompted Turk refusal (as Makarios probably knew and expected it would) since the Turks saw the veto as a way for them to preserve their power as a minority. The Turkish Cypriot leader, Dr. Kucuk, consulted with Turkey on the proposal, then rejected it, saying it was an attempt to undermine the constitution.

Tensions continued to rise. On December 21, 1963, Nicosia experienced huge amounts of violence when the EOKA and other Greek guerrilla forces attacked civilian Turkish Cypriots in response to anti-Greek rioting over the proposed changes/tensions, and the resultant fighting lasted until April of 1964, by which time many Turks had been forced to leave government positions and move away to their own enclaves of Turkish communities. While the US and Britain proposed partitioning Cyprus and allowing each community to join up with its respective mainland, Makarios refused.

UN peacekeeping forces had been approved by the UNSC in March of 1964, authorized to be there for 3 months. They ended up staying until past 2001.

However, in 1967, the crisis erupted again (after relative calm until then). A nationalist military coup in Greece spilled over into Cyprus, as 12,000 Greek troops appeared on the island, and the leader of the EOKA (Grivas) became more active on the island again. The Turks were the first to fire in the tense atmosphere, but Grivas responded massively, and the resultant violence led Turkey to threaten war against Greece. Intense international negotiations led to Grivas and the troops withdrawing, and a cessation of violence...for the time being. However, in 1967 the threat of Turkish invasion caused Makarios to reconsider the idea of unification with Greece; this would prove to be a huge issue in the 1974 crisis.

Cyprus - 1974, and the Invasion by Turkey

Here's where things get really, really hairy. Cyprus managed to semi-function, limping along, until 1974. Makarios was no longer in favor of Greek unification, and while tensions remained high, there weren't any really big issues to note that I've read about.

Until July 15, 1974, that is.

The EOKA, on July 15, 1974, initiated a coup against Makarios with the support of Greece, and succeeded handily. Makarios left the island, and they took power. The Turks viewed Greece's involvement as proof that they broke the 1960 London Agreement that established Cyprus as an independent republic. The "Treaty of Guarantee" of 1960 said that the British would intervene if necessary to enforce the London agreement (well, it called for "consultations" between all the parties again, but it was pretty clear what it meant). It also said that if the British didn't, that any of the parties could act to remedy the problem themselves. And remedy they did.

The Turks invaded on July 20, 1974, after consulting with the British (who basically refused to get involved). Three days later, the Greek military dictatorship in Athens collapsed (meaning that there was no longer really a threat of war between Turkey and Greece, both NATO members), but Turkish troops were now in Cyprus. And while one could argue that their intervention was justified, was the occupation?

Well, the US (specifically Kissinger) is accused of having told Turkey that they could reasonably occupy about 1/3 of the island. There's reason to believe the British and US both knew of the invasions before they came, and either said nothing or condoned them. And while it's possible to say the intervention was allowed, there's almost no one I have read who thinks that it was justified to continue occupying the area (which is why no one besides Turkey has recognized "North Cyprus"). Due to the lack of real opposition, Turkey was able to grab that 1/3, though, with very little trouble. Despite talks affirming commitment to Cyprus independence, the fighting mostly continued until the 16th of August, when the Turks finished grabbing around 36% of the island, and at which point things remained.

Please ask questions if you have them! If I was vague it's because I haven't slept much and because of the character limit.

Sources:

Application of Cyprus to the European Union and the Cyprus Problem Kamil Sertoglu and Ilhan Ozturk Emerging Markets Finance & Trade, Vol. 39, No. 6 (Nov. - Dec., 2003), pp. 54-70

Cyprus, Britain, the USA, Turkey and Greece in 1977: Critical Submission or Submissive Criticism? William Mallinson Journal of Contemporary History, Vol. 44, No. 4 (Oct., 2009), pp. 737-752

The European Union and the Cyprus Problem 1961-2003 Meltem Müftüler-Bac and Aylin Güney Middle Eastern Studies, Vol. 41, No. 2 (Mar., 2005), pp. 281-293