While I agree on most of what /u/Raventhefuhrer says, I'd like to disagree on a functioning nation - for most of the European colonial era in Africa, Ethiopia was split in a series of feuding feudal or warlord states in what was called Zemena Mesafint, or the Era of Princes. The Emperors were then mere figureheads, not unlike the Japanese Emperors forced to accept shoguns as the real wielders of power.

One also needs to remember that Europeans advancing deep into Africa is a rather young phenomena.I'd like to argue that it was not until the Gatling gun (and then machine gun) and synthetic quinine (1866 and 1850 respecivtely) became widely available that Europeans could really make inroads into Africa south of Sahara.

Ethiopia is mountainous and very inaccessible. Geography alone gave the Ehtiopians some protection, especially as the Nile is not navigable above the (now submerged) Wadi Haifa in southern Egypt since it flows from a number of rapids from Khartoum. To further complicte things, the Blue Nile (which originates in Ethiopia) is not navigable to Lake Tana, since the massive Tis Issat falls come just below Lake Tana. The Blue Nile also cuts a deep canyon that separates the north of the country from the south.

To be short, before modern machinery, it is hard to get to Ethiopia.

Immediately south of Ethiopia, which is the direction that Europeans would come from before the creaton of the Suez canal 1869 is the realm of the tsetse fly which carry sleeping sickness (or trypanosomiasis for animals), which makes it nearly impossible to bring livestock, including horses, mules and donkeys from that direction, which would be required to bring supplies for any kind of invasion.

As you can see, geography and fauna conspire to build a natural defence around Ethiopia during the colonial age.

Adding to this is a belt of muslim states around Ethiopia, which Ethiopia often feuded with, but all had contacts with the Arab world, especially the Omani Empire and Egypt and had access to high-quality steel weapons and other modern technology. Feuding with these states forced Ethiopian warlords and princes to adapt to new technology and tactics.

Another benefit for Ethiopia was the European Prester John myth. During the Crusades and then at the height of the Ottoman Empire, a myth of a christian nation on the other side of the muslim world, that would arise to help christian Europe was widely circulated. Prester John's Kingdom was placed alternatively in India and in Africa. There had been sporadic contacts between the Empire of Ethiopia and Europe during the Medieval Era, which fed these myths.

When the Portuguese established contact with the Ethiopian Empire in the late 15th and early 16ht centuries, they thought they had found the legendary Kingdom of Prester John, but in reality they had found a country beluegred on all sides by muslim states. However, Portuguese technology and weapons (such as gunpowder, cannons and arquebusiers) quickly made their way to Ethiopia and let the country reverse many of the losses against the coastal Sultanates.

Once the Europeans arrived in the area in force in the 1880s (after the conference of Berlin 1878) the Ethiopians had a long tradition of civil war and dealing with their coastal muslim enemies. However, they were more advanced than many of the more tribal in the interior of Africa to the south. For example, the Ethiopians had an entirely coin-based economy, based mostly off Austrain Maria Theresa silver thalers (it is estimated that as much as 20% of the more than 360 million thalers that were minted ended up in Ethiopia).

The Italians occupied Eritrea and southern Somalia, but still felt cheated as other European nations had gained much more territoy in Africa and thus wanted to join their two colonies by occupying Ethiopia.

The Treaty of Wuhale from 1889 between Italy and Ethiopia contained the phrase "the Emperor can use the Italian foreign office in his relations with other powers" in the Amharic version, and "the Emperor must use the Italian foreign office in his relations with other powers", effectively making Ethiopia an Italian client and protectorate. It is debated wether the Ethiopian version had can as Menelik claimed, or must as the Italians claimed - it is possible that Menelik signed the treaty well aware of its contents but intending to reject it later, when he was strong enough. The Italians gifted Menelik money and some modern arms, which he used to bring the outer part of the Ethiopian Empire (primarily Oromo and other nomad people) under his direct control. Thus able to tax and force tribute out of them, he used that money and the money from a series of bountiful harvests to buy modern arms.

When things came to head, the Italians claimed that Menelik had signed an Amharic version containing must but had edited that part out of his document, while Menelik rejected that notion. Italy then went to war in the First Italo-Ethiopian War and was severly trounced at the Battle of Adwa 1896.

The Italians had brought about 17 000 men to the battle, with 56 70mm M1885 mountain guns. They calculated that the Ethiopians would field about 40-50 000 men, of which 1/4 would have modern rifles, 1/4 antiquated rifles and 1/2 spears and swords. The Ethiopians brought about 120 000 men to the battle, of which about 1/2 had modern rifles, 1/4 antiquated rifles and 1/4 spears or swords. The Ehtiopians has also purchased modern 50 Russian 63,5mm M1883 mountain guns that actually outranged the Italian artillery.

With the defeat of the Italians at Adwa, most of Europe saw the Ethiopians as if not equals, at least better than other African people. I have read accounts of "racial researchers" claiming that the Ethiopians were "dark caucausians" and not "negroes" (their terminology, not mine) since "negroes" could not defeat caucausians.

it is essentially the only country which avoided colonization by Europeans

This is incorrect. It is simply the only African country that survived colonialism intact. (if we consider Liberia to be a colony of African American immigrants reigning over the local population - you might be interested in this Slate article)

For example, Siam, now Thailand; it escaped direct colonialism, partly through the modernizing efforts of the monarchs of the Chakri dynasty (especially King Mongkut, the inspiration for the musical The King and I) and partly because the British in Burma and the French in the east wanted a buffer zone in between each other. Nevertheless, Siam conceded its peripheries to France and Britain - Cambodia and Laos to the former, the northern Malay states to the latter - and much of the kingdom fell under the hegemony of the French or the British. It was forced to accept the extraterritorial rights of colonial powers and fixed custom rates. Still, the country retained its independence, and the current king, Bhumibol Adulyadej, is the ninth Chakri monarch.

Japan was also never directly colonized; in fact, it modernized (or Westernized) very capably to become a colonial power itself, annexing the Ryukyu kingdom in 1879, defeating China in 1894 and Russia in 1905, and finally directly annexing Korea in 1910. Thus, Korea was never colonized by Europeans but by the Japanese Empire, although Russia had influence there in the late 1890s and early 1900s and Japanese colonialism wasn't overly different from that of contemporaneous European powers. The Making of Modern Japan by Marius Jansen might be worth checking out if you're curious about modern Japanese history, or perhaps Inventing Japan: 1853-1964 by Ian Buruma.

This thread from a few years ago has a top level answer.