Well I have an answer for you, but be warned, it's pretty heavy on monetary economics terminology. I don't know how much you already know about financial markets, and I've tried to explain some of the mechanisms, but feel free to ask questions if you don't understand some of the terminology I use. 

Bretton-Woods was a system of monetary management started by the Allied nations during WWII, though the Soviet Union didn't ratify the final accord and therefore wasn't part of the system. Some parts of the system are still in effect - the IMF was established by the Bretton Woods accord. The accord also established the International Bank for Reconstruction and Development, which is now part of the World Bank. It didn't include a trade aggreement, but the General Agreement on Trade and Tariffs (GATT), the pre-runner to the WTO, can be traced back in part to its ideas. But when people talk about Bretton Woods falling apart, they are generally referring to the system of fixed exchange rates the accord set up, and I'll discuss that for the rest of this answer. The Bretton-Woods exchange rate system effectively ended in 1971 when the USA ended convertibility of the US dollar to gold (the "Nixon shock"). 

The period between WWI and WWII had seen many monetary problems, eg the UK's struggle to return to the gold standard in the 1920s, or in the 1930s, rounds of competitive devaluations - when one country would lower their exchange rate, favouring their own exporters - at the cost of their consumers and that of exporters in other countries. This is known as "beggar your neighbour" policies.  

The idea of Bretton Woods was to avoid the inter-war experiences with an international system of tight management of exchange rates and capital flows, particularly speculative flows. This was expected to promote international trade and help in maintaining price stability. This fixed system and tight management was consistent with the intellectual current of the time which favoured governments strongly managing their economies, driven in part by experiences of the Great Depression but also I think partly from the way in which neoclassical economics was (and mostly is) still taught, with its emphasis on market failures and neglect of government failures. 

The USA was the main player in the Bretton-Woods system. Other participating countries included Western Europeans, Canada, Australian, NZ and Japan, but I'm going to copy my macroeconomics professor and refer to the others as "Europeans" for brevity. The western Europeans of UK, Italy, West Germany and France were the major other economies in the system for most of the period, my apologies to Australia, Japan, etc. 

Under the Bretton-Woods currency system, the US was obliged to maintain the US dollar value of gold at 35 dollars to an ounce - so the US was on a gold standard. The Europeans were obliged to peg their currencies to the US dollar. The pegs could be changed, but a change of more than 10% required approval from the IMF (so of course countries would change their pegs a bit less than 10%). 

The Bretton Woods is generally accepted to have ended for two, interlinked, reasons:

 * The USA for a long period ran a balance of payment deficits (in other words, the USA was importing more goods and services than it was exporting, and thus was borrowing money to pay for its exports), placing pressure on its ability to maintain the gold peg: foreign-held external dollar liabilities exceeded US gold reserves by 1960.  

 * The pegging of Europeans' currencies to the US dollar meant that when the USA boosted inflation in the later half of the 1960s (to pay for the Vietnam war and expanded social programmes), the US inflation passed on to them. This was politically costly for governments in other countries where the population opposed inflation, especially West Germany and Japan. And of course West Germany and Japan's economies were much larger at the end of the 1960s than they were at the start of the 1950s, so they were much more influential in terms of trade and currency flows. 

Currency traders may not be wise, but they are smart, and can read political tea leaves as well as any. Between these two factors the financial markets started to expect devaluations. If you expect a currency to devalue, the way to make money is to sell that currency now, with the plan to rebuy the currency more cheaply once the devaluation has taken place. Under a system of fixed exchange rates, if someone is selling currency then the central bank is obliged to purchase the currency in exchange for something else (under Bretton-Woods that was US dollars, or gold). Clearly if everyone is trying to sell currency then the central bank faces running out of the something else, creating pressure for a devaluation. Therefore expecting a devaluation can become a self-fulfilling prophecy. 

A government can seek to defend its currency by placing limits on how much of its currency it will purchase. This however makes life hard for importers - both consumers and also businesses seeking imports as inputs to domestic production and/or exports. The central bank can try to let select groups have preferential access to foreign currency, but this involves high monitoring costs and creates incentives for people to try to find ways around exchange rules, creating yet more monitoring costs. Furthermore, these types of regulations discourage exporters (or other residents) with foreign exchange from exchanging that for the domestic currency, as the exporters can't be confident that they will be able to reacquire foreign currency in the future, thus exacerbating the problems of a shortage of foreign exchange. 

Fukumoto(2011) also argues that there were continued and growing international price differences between many of the participating countries, and that these would have made Bretton Woods unstable, even without the worldwide inflation of the latter half of the 1960s.  

Eventually the domestic costs to the member countries, both economically and politically, of maintaining Bretton-Woods got too high. West Germany left Bretton-Woods in May 1971, European countries started demanding more and more redemption of their dollars for gold, Switzerland left on 9 August, and Nixon announced the US was abandoning gold convertibility on 15 August 1971. Basically the system collapsed. 

There were other measures that the USA and the Europeans could have taken to maintain the Bretton-Wood system. The US government could have raised taxes to pay for the Vietnam war, but there wasn't the political willpower to do that. The Europeans could have paid more of the costs of the Vietnam War, (or US government expenditures more generally) but there wasn't the political willpower to do that either. And of course there are strong governance reasons for the Europeans to not do that.

The USA did try capital controls, eg tightening the limits on the value of duty-free goods, and imposing an Interest Equalisation Tax, but this was costly to the US economy and not sufficient for the scale of the problem.

The Bretton-Wood system was an illustration of how important choices of policy levers are, if you get them wrong it's easy to get into a spiral where the more you try to control things the more you need to control and the more and more rigidities you collect and the more fragile you make the whole system.

I know I've not addressed your question about what came after but I'm having access issues with some journals right now, I hope to get to it this evening but it may be delayed until after Easter. 

Sources:

Anthony M. Endres & Alan J. Rogers (2014) Trade Policy and International Finance in the Bretton Woods Era: a Doctrinal Perspective with Reference to Australia and New Zealand, History of Economics Review, 59:1, 62-81, DOI: 10.1080/18386318.2014.11681256 - ungated https://www.researchgate.net/profile/Anthony_Endres2/publication/264196643_Trade_Policy_and_International_Finance_in_the_Bretton_Woods_Era_A_Doctrinal_Perspective_with_Reference_to_Australian_and_New_Zealand/links/553e06180cf2c415bb0f8f7a.pdf

Yukio Fukumoto (2011), Another Look at the Underlying Cause of the End of the Bretton Woods System: International Price Differences Perspective, Review of International Economics, 19(5), 852–864, DOI:10.1111/j.1467-9396.2011.00990.x

Post Bretton-Woods collapse

Well the journal article I was looking for wasn't relevant to post-Bretton Woods  anyway. C'est la vie.

The Nixon Shock in August 1971 was followed by the December 1971 Smithsonian agreement. This agreement set the gold price at US$38 per ounce and other key countries (the G-10, the main Western European countries + Canada and Japan) re-set their exchange rate pegs at a higher rate to the US$. This agreement did nothing to fix the fundamental imbalances, the US dollar was still over-valued relative to the Europeans. The UK decided to let the sterling float in June 1972 under severe pressure from speculators. In the US, politicians became increasingly distracted from international monetary arrangements by the Watergate scandal. The Europeans strengthened controls on capital flows but this was costly, and the Danes decided to follow the Brits and let their currency float. The Smithsonian agreement finally fell apart entirely February to March 1973. Japan floated its currency in February. The 6 European Community members of the G-10 tied their currencies together and let them float against the US dollar. 

Since then there's been a mixed system, some countries float their currencies, some countries peg (which can be done unilaterally if a country is willing to basically hand over its inflation rate to whatever country it pegs to). Some countries have a "dirty float" - floating but the central bank sometimes intervenes. The IMF reworked itself as a lender to developing countries and a general busy-body. And of course the World Bank is still going strong.

So that's a summary of the history of Bretton Woods. Hope the jargon wasn't too snooze-worthy.

Source

https://www.federalreservehistory.org/essays/smithsonian_agreement