It is significant for many reasons--heading out the door, so I must be briefer than I would want.

This was a foreign intervention into an independent (at least in theory) country by fellow Warsaw Pact Nations. The Warsaw Pact, or the socialist NATO, was signed in 1955, and was a military alliance, but was more importantly, an official means for the USSR to assert political control over these nations as the defenders of the revolution, and stated they could intervene to defend the revolution. It was the formal declaration of the Breshnev Doctorine, which states: " Eastern Bloc When forces that are hostile to socialism try to turn the development of some socialist country towards capitalism, it becomes not only a problem of the country concerned, but a common problem and concern of all socialist countries." Hungary was the first trial run of this policy enacted.

It also became the first test of the Truman Doctorine in Europe that actually came to actual shots fired. There were a few instances in Germany that came dangerously close to actual war, but never quite got there. This was time for the US to actually decide if they would go to war to defend democracy. They didn't. Some of the resistance were on the radio begging for US/NATO intervention, as they could only hold out for a little longer.

The Soviets also broke a declared cease-fire when they crushed the revolution. There were documented instances of torture, mass arrests, and a lot of refugees as well as quite a few deaths. It's here that the Soviets appear as the 'evil empire' and the idea that any of the satellite states have any true autonomy is firmly defeated.

It was also a matter of bad timing. The Suez crisis happened at the same time. It was noted that the US could not defend Britain and France's 'intervention' in Egypt while condemming the USSR for intervening with a country it had a political pact with.

It's clear the neither US and USSR don't want to fight a war in Europe. When the dust settles in 56, Europe has become a stalemate. Neither WP or NATO forces want to wage a war in Europe, and both sides have their own sphere of influence. The true cold war will be fought by proxy in Africa and Asia.

Here is an NSA brief on the 1956 uprising.

Ok, so my area of expertise is the Middle East, so you're going to have to bear with me for a bit because its going to look like I'm replying to the wrong thread, but I promise you this will tie in to Hungary, but isn't strictly about Hungary. Hopefully someone with more direct knowledge of the Cold War can speak to that revolution itself, but I think this will contribute to an answer.

In terms of the history of the cold war, there were two "big events" in 1956, and both occurred near the end of the year. As you mention, the first and possibly better known is the Hungarian Uprising. The second was the Suez Crisis. Ultimately their impact upon one another was a pivotal moment in the cold war.

Gamal Abdel Nasser came to power in Egypt as a result of the Egyptian Revolution in 1952. It was essentially a military coup with massive popular backing led by a group calling itself the Free Officers Movement and led by Nasser (Nasser wasn't initially made president, but Mohammed Naguib was basically a figurehead.)

In the early days of the Cold War when Europe and the major powers were taking sides or otherwise being compelled to join sides Nasser was a bit of an oddity. His economic policies leaned towards big state socialist-style politics but without any of the Marxism and certainly with none of Communism's internationalist aspirations (edit: although of course he did very specifically have a pan-Arab ideology.) Nasser and Nasserism were decidedly Egyptian. He would later be a founder and leading figure of the "Non-Aligned Movement."

The focus of his policies was the construction of a massive dam on the Nile River at Aswan. Initially, in a bid to get him on side against the Russians, the US backed Nasser. But when Nasser signed an arms deal through the Czechs and recognized Communist China, the US withdrew its pledge for funding.

Nasser then signed a funding agreement with the USSR. To pay for the rest Nasser then nationalized the Suez Canal, which until that point had had joint ownership with Great Britain. This is in July of 1956.

The nationalization of the Suez, the receipt of Czech weapons, and the backing of the Soviet Union set off alarm bells in London, Paris, and Jerusalem. Anthony Eden essentially came to the conclusion that Nasser was an Arab Hitler and this was his equivalent of annexing the Sudetenland. France and Israel were extremely close allies at this point, much more so than the US, which actually had an arms embargo against Israel. Having just fought against Egypt for its independence 7 years earlier, Israel viewed the Egyptian actions as an existential threat.

The three countries convened a secret meeting at Sevres, and plotted (yes, literally plotted, this whole scheme reads like a bad spy novel from this point on) to have Israel start a pre-emptive war against Egypt, and then Britain and France would intervene under the guise of "peacekeeping," establishing an armistice line of British and French occupation conveniently requiring them to occupy the Suez Canal Zone. The meeting at Sevres started on October 22nd and concluded on the 24th.

(Now back to Hungary!)

Literally as that meeting is taking place, on October 23rd, 1956, the first mass protests in Hungary against the Soviets break out. Again, I'm not an expert in this history, particularly on the Hungarian/Soviet side, but my understanding is that in the White House and the in the State Department, this was viewed as a huge opportunity to crack Soviet domination of Eastern Europe. That this was an enormous moment to show the world the evil oppression of communism.

The Israelis invaded Egypt on the 29th of October. Almost from the start it was laughably obvious that the whole thing was an outside conspiracy between France, Britain and Israel.

The Eisenhower administration was furious. They couldn't understand why Britain, in particular, would mount such a perilous invasion to secure a canal that went to an empire that Britain no longer held.

It also completely comprised the American negotiating position vis a vis Hungary. Rather than a clear black and white case of Soviet aggression in contrast with Western benevolence, the Soviet actions in Hungary were evenly matched by the brazen display of Western imperialism over Suez.

The Eisenhower administration decided it had to end the conflict immediately to save what face it had left. The US threatened to dump its Sterling reserves which would have instantly bankrupted the UK and left it unable to afford its own food and fuel imports.

The end result, and to try to bring this back to your question, is that the Soviets were ultimately allowed to have their way with Hungary. At the same time Anthony Eden announced a unilateral ceasefire, the Soviets were crushing the last holdouts of the Hungarian resistance in the first week of November, 1956.

The United States emerged as the uncontested dominant force in the western world, with the power and the will to force former great powers like France and the UK to heel. But ironically, rather than doing so in triumph as a world leader against soviet aggression in Hungary, the US assumed this role as a result of the massive western debacle at Suez.

So the TL;DR as I understand it is that the Hungarian was a huge own goal by Western powers because of the Suez Crisis that greatly Exacerbated the Cold War.

Sources: For Eden's views on Nasser, Troublesome Young Men by Lynne Olson, for the Suez Crisis itself see D.K. Fieldhouse's Western Imperialism in the Middle East and for the Cold War in the Middle East and the Suez generally see Eugene Rogan's The Arabs: A History The latter two discuss Hungary in the terms that I've used here.

/u/CptBuck and /u/Lina1611's answers are both good, and the former also lays out the international aspect which is very important context for the period, but there was another enormous significance to Hungary which was the topic of a book I recently reviewed and really interesting. (CC: /u/prx124, this hopefully answers your [self-answered] question in a little more detail).

Hungary, in many senses, was the culmination of the US's attempts at 'political warfare'. It was an interesting era, when 'containment' as a strategy was being phased out (for all the wrong reasons: democratic pressures to be seen to 'do' something; a naive military that thought it had all the answers; irresponsibly policymakers who thought they could get big results without spending much blood and treasure; the list goes on) and replaced with 'rollback'—the strategy to pressure the USSR and attempt to break small parts off at a time by undermining Communist regimes (and endorsing anti-Communists over non-Communists, seemingly irrespective of feasibility) in the way the Soviets had tried (with little success) to undermine democracy our side of the Iron Curtain.

There had been a couple of 'successes' (very important not to look beyond a few years, or they start looking like abysmal failures inflicting grievous suffering for almost no reason at all) in Guatemala and the coup in Iran, and this convinced the administration that CIA-run 'political warfare' was the cost-effective way forward. Sadly, they failed to note that Guatamala was only a success because the threat of US intervention sat squarely behind it. When similar operations were attempted in Albania (the smallest and weakest of the Soviet bloc) as well as Ukraine, Poland, Yugoslavia and elsewhere they were hopelessly unsuccessful and usually undermined US policy elsewhere (the State Department was actually trying to make friend with Tito, whilst the CIA was trying to overthrow him; in Ukraine, the CIA was backing the only force that the Nazis didn't have to pay to kill Jews throughout WWII; in Poland, the CIA fell for an obvious KGB/UB ploy, thus teaching the Soviets everything about their operations; and in Albania their agents were rolled up and shot, publicly, affirming to the world the USSR's narrative of US imperialism gone wild).

Hungary was the big culmination. Poland had just been granted some concessions by Khrushchev, who was in the process of de-Stalinisation and active reform of the USSR and its external relations. The US response, through Radio Free Europe (staffed mostly by bitter Hungarian dissidents, but funded by the CIA), was to make Hungarians and others aware through mass radio and other propaganda operations. This was pretty hot in Hungary, especially amongst young students, who protested, demanding similar concessions. Then things got out of hand. Protests were fired upon, whilst RFE started promising American assistance (that was never on the table), instructing protesters how to destroy railway lines and ransack barracks for weapons. Eventually a deal was put together that would grant some concessions, provided Nagy remained in power. RFE made clear its refusal, protesters did likewise, and as a result Nagy radicalised and started talking about pulling out of the Warsaw Pact. Soviet intervention ensued and, of course, the protesters never stood a chance and were massacred, whilst the West stood idly by, knowing full well it bore some responsibility for the bloodshed yet could do nothing to stop it now. Of course, military leaders were calling for the US to go all in (as is their wont, if you read your US history). Eisenhower's retort to Charles Jackson is pretty good on this one (and bears serious relevance today):

[Extreme paraphrasing:] "I know your whole being cries out for action on the Hungary issue, but to utterly destroy Hungary, should it become scene to a bitter conflict, is in no way to help her."

Political warfare stopped being quite so popular soon after Hungary, and many of George Kennan's principles were re-adopted to replace the hurried mistake that rollback was. Nevertheless, (and here's where I depart from source-based history) at some point it became wildly popular—even if just in its out-sourced RFE-like guise, as we see so much of today—because sooner or later we start seeing CIA operations spanning the globe again and achieving a similar success rate to the period covered in the book (next to none, in fact, negative if that's possible). I digress. If this is interesting (it's intelligence ops, of course it's interesting) then that's likely a book for you. I'd link my review (in which I draw all sorts of parallels to modern political warfare, the likes of which we're seeing in Ukraine), but it's yet to be moderated, and I'd rather avoid a) posting something crap, and b) triggering the automatic plagiarism program.

Book: Long, S., 2014. The CIA and the Soviet Bloc: Political Warfare, the Origins of the CIA and Countering Communism in Europe. London: I.B. Tauris.

As many of the other posters have mentioned, the fact that the Hungarian revolution happened at the same time as the Suez Crisis boded badly for the Hungarians, as the Western promises of aid came to nothing. It is interesting to note, however, that Canada actually raised a brigade of volunteers to send to Hungary.

I would rather focus on the long term effects of the revolution, but to do this, you need to understand the situation in Hungary beforehand. Before the revolution, Hungary was brewing with dissent, and most people refused to believe that the Soviets were there to stay. The revolution itself had many socialist leaders who disagreed with the authoritarian Soviet leadership. After the revolution however, the crackdown on dissent was extremely severe. An entire complex on Andrássy Street (now a museum) was dedicated to capturing, interrogating, imprisoning and executing revolutionaries. Essentially, any and all dissent against the Soviets was quashed, and Hungarians realized that the Russians weren't going anywhere for a while. Post-cold-war the revolution became a Hungarian national holiday, and received much national focus. In both Hungary and the rest of the world, the Soviet Union's crushing of the revolution firmly established them as the "bad guys".

Another important thing to note about the revolution, is that it contributed to a long-term easing of the harshness of the regime. Once the secret police had imprisoned or killed any and all dissenters, the Soviets actually resorted to appeasement to prevent another such uprising. Whether this is directly a result of the revolution is not for sure, however it is true that exponentially more workers medals were given out afterwards, and the period of " Goulash Communism" followed not long after as well. Hungarians thus like to refer to the revolution as "the first chink in the Iron Curtain."

Sources: -Hungarian high school history -Terror Ház -My 4 grandparents are '56ers

Edit: Additional Sources:

-The Hungarian Revolution and War for Independence by Lee Congdon, Bela K Kiraly, Karoly Nagy (English)

-1956 Sajtoja by Tudositasok Kiada (Hungarian)

This is meant to a supplement to the great stuff already posted. I'm drawing here mostly from Tony Judt's wonderful "Postwar". The Hungarian Uprising, and more specifically the violence of the Soviet crackdown, proved to a major moment in the thinking of many of the Left in Western Europe, as well as some in the Bloc states. Up until then the thinking was that the purity of Marxism-Leninism still existed and that Stalin had just been a brutal perversion, now corrected with his death in '53. With the Hungarian Uprising the Soviets showed that the foundation of the Soviet system was physical power, not idealogical purity, and that the system may in fact be at fault, not just Stalin. After all, the main reason that the Soviets cracked down was not because of policy difference, but on the Hungarian pretense of being able to define Communism in their own way a la Tito's Yugoslavia. All of this led to significant disillusionment among the "faithful" Left who still supported the Soviet Union in principle.

Followup question. I have read that the main demands of the revolution were not the end of "socialism" or the privatization of the economy and return to capitalism, but rather democratic control over the nationalized planned economy. Of course, I would understand if elements wanting a return to capitalism were also involved in the melee. But my question is, which was dominant in the movement?