So, /u/YesThatLioness is basically correct. "Allies" and "Axis" were terms that were known to both sides before the war and that were generally used by both sides during the war, which is in itself interesting because obviously both sides had positive connotations for their own term and negative ones for the other.

So, the "Axis powers" was a term defined by Benito Mussolini and comes from Italian and he actually used it the first time for Italo-German relations in 1923 when he was trying to secure political support from the Weimar Republic against Yugoslavia and Belgrade's French ally during the Italo-Yugoslav Fiume Crisis. He specifically said that the axis of European history passed through Rome and Berlin, forgive my potentially poor Italian translation. Mussolini had had Machiavellian-grounded pro-German sentiment basically since after the end of World War 1, as, although Mussolini had been a veteran of Italy's Entente WW1 army, he saw a rivalry with Italy's former allies, especially France, as inevitable. France immediately after World War 1 began securing an Eastern European counter weight network of allies against a potential new German or Soviet threat. France during that time politically allied Poland, Czechoslovakia, Romania and, critically to Mussolini, Yugoslavia.

Well, fast forward a decade. Germany was now a new threat under one Adolf Hitler, and Mussolini was initially very hostile to the new fascist on the continent. The conflict between Germany and Italy pertained specifically to "the Austrian Question", i.e. the future of Austria as an independent state. Hitler, Austrian by birth and violent German nationalist, left no question between his Mein Kampf and his daily political rhetoric, that he wanted a Greater German union between Germany and Austria, upsetting the balance of power in central and eastern Europe in Germany's favor and thus disadvantaging Italy. Mussolini thus tried to realign with Britain and even France in the "Stresa Front", where the three countries affirmed the Locarno Treaties in which, in exchange for German accession to the League of Nations, Germany had promised to respect the western border in 1925. With this reaffirmation in 1935, it could have been very possible that Italy as the protector of Austria and even Belgium and the Netherlands could have found itself as an Allied Power in World War 2 as well. But then the UK undermined the Stresa Front with the Anglo-German Naval Treaty and, to add insult to injury, the Anglo-French alliance with Italy immediately turned to distrust when Italy invaded independent Abyssinia (Ethiopia) in late 1935.

Thus, Italy once more realigned, this time firmly into Nazi Germany's camp. On 1 November 1936, Mussolini made the famous speech declaring a Asse Berlino-Roma, 'Axis Berlin-Rome', around which 'the continent would rotate'. Later, this Axis term, also used in German as 'Achse Berlin-Rom', would be expanded to include another city - Tokyo. In Italian literature, you will sometimes find it referred to as The RoBerTo Axis, Rome-Berlin-Tokyo.

Now, the term "Allies" is somewhat easier. "Allied Powers" had been the English-language term of choice for the very French-sounding Entente of World War 1, and "Allied Powers" as a phrase was frequently used in post-war documents and treaties, making it a derogatory term for the likes of Adolf Hitler. Thus, going into World War 2, the Allies just retained their English war from the last war, even though of course the name Entente had fallen out of favor after the end of the French-Russian Alliance of before World War 1.