In many ways, it can be argued that WW2 was a continuation of some of the issues that caused, and were caused by, the First World War.

Perhaps the biggest reason was the Treaty of Versailles, the peace treaty that ended WW1. In the treaty, it was made clear that the Western Powers blamed Germany for the war, since they 'needlessly' attacked Belgium/France (enacting the Schlieffen Plan) after Austro-Hungary's declaration of war against Serbia (and Russia by proxy). As such, much of the treaty was designed at limiting Germany's power to ensure another war would not happen.

Germany was therefore forced to pay 132 Billion Marks ($33 Billion) to repair France after the war, in lieu of the catastrophic damage caused to both the civilians and state's property. This is a figure that effectively bankrupted the country, and the issue was only exacerbated further once the Great Depression hit in the late 1920s. The Great Depression plunged Germany into an economic crisis, wherein hyperinflation caused prices of staple foods to skyrocket. For example, the price of bread was said to be changing so rapidly that it would be 1 million marks in the morning and 2 million marks in the evening. There are also multiple cartoons depicting this stage of Germany with people taking literal wheelbarrows of cash to buy the most basic of products. I won't go into it too much here, but the poor state of the economy was a large reason for Hitler's rise to power as it caused a polarisation in German politics forcing many people to choose between either fascism and communism.

This was not the only thing the Treaty forced onto Germany however. They were also massively limited militarily in order to encourage international disarmament (and because France was incredibly bitter). They were no longer allowed paratroopers in their forces, they were forced to demilitarise the Rhineland (bordering France), and they were also limited to six pre-dreadnought battleships, six light cruisers, twelve destroyers, twelve torpedo boats, and absolutely no submarines. Moreover, Germany was forbidden to have an air-force, and their army numbers were reduced to a meagre 100,000. To put this all into a little bit of context, it is important to note that Germans were an incredibly proud and militaristic people. All of these reductions and concessions were a national embarrassment and the acceptance of these terms by their government was seen as shameful and humiliating.

Lastly, Germany also lost 25,000 square miles worth of territory (not including all of their overseas territories which were given to other European powers) and 7 million people. This was 14% of its landmass and 12.5% of its people.

Why does all of this matter? Hitler. As a soldier during the war, he in particular was ashamed of his government's acceptance of the treaty, and made it his mission to make Germany great again. He was able to capitalise on this feeling that the German people had by promising to reverse the Treaty. Slowly but surely, once Hitler gained control of Germany in 1933, he ripped apart the Treaty of Versailles.

The first significant part of the Treaty that Hitler ignored was the remilitarisation of the Rhineland in 1936 (Germany had already been secretly rearming basically since the treaty was signed), and this was largely tolerated by the likes of Britain because they felt the treaty was overly harsh to Germany. Two years later in 1938, Germany then annexed Austria, which again was tolerated for similar reasons. That same year, Hitler turned his eyes to Czechoslovakia, where a significant portion of Germany's land was stripped away (that also was a strategically significant area due to natural mountain defenses as well). This culminated in the now infamous Munich Agreement where Neville Chamberlain (British PM) enacted his policy of appeasement which allowed Hitler to take the area known as the Sudetenland. Part of this agreement however was that the rest of Czechoslovakia wouldn't be taken and that would be the end of German expansion. Hitler agreed expansion would end and led to Chamberlain's 'peace in our time' speech.

The next year (1939), Hitler seized the remaining areas of Czechoslovakia, and it was now clear another war was going to start. Britain and the other allies warned Hitler once more to curb his policy of Lebensraum (living space in the East) and that war would be declared if he attacks Poland (his next likely target). Obviously, Hitler ignored this (thinking he would be appeased once more), and seized Poland in September 1939, at which point WW2 had begun.

I've skipped over A LOT of other factors in this overview, although it should give you a relatively good idea as to why the wars were so close. The main things I've missed are the failure of the League of Nations (a precursor to the UN which the US never joined- the main reason it failed), Japanese aggression in the Pacific, and the Abyssinian Crisis.

For the second part of your question the reasons are many, and can likely be explained much better by someone with some expertise in the Cold War or International Relations, but suffice it to say that the UN (League of Nations 2.0) has been far more effective at peacekeeping through its use of sanctions etc. Another main reason is the creation of nukes, which led to the policy of MAD (Mutually Assured Destruction) between the major powers of the world. The idea is basically that if war breaks out between two nuclear powers (e.g. US and USSR), they both have such huge nuclear stockpiles that the world would end as we know it. Because of this, neither declares war. As for Europe, the EU has done wonders for international diplomacy. It seems that most nations have realised that the world is not a zero-sum game and that international cooperation is a much greater way to achieve success than simply destroying your neighbours and stealing their stuff. The integration of their economies has meant there has been very little reason to declare war, and they also remember the horrors of the two world wars that have passed.

The last reason there hasn't been a world war since is a slightly contentious issue which argues that democracies simply make war less likely. To date, there have been no major wars between two democratic societies, and I would argue that this is because democratic societies tend to be much less authoritarian/nationalistic and therefore more likely to focus on trade rather than war, but that's just my opinion.

Sources:

P. M. H. Bell, The Origins of the Second World War in Europe

Weinberg, Gerhard. Hitler's Foreign Policy 1933–1939: The Road to World War II

Bruce Russett, Grasping the Democratic Peace

Dan Smith, The State of the World Atlas