I've done a small amount of research into it, and my impression is that it was a relatively quiet, very gradual process that occurred over several centuries. However I find it a little difficult to get a succinct, coherent rundown of the events that lead from an absolute monarchy to the system we know today. So what were the major landmarks and/or driving events that led the monarchy to relinquish power?
The most important point to understand about democracy in the United Kingdom is that it was granted rather than taken, but not in the way you might expect. The seemingly obvious answer is that kings gave power over to the people, but this is, as any British historian will tell you, not really correct. Rather, it was Parliament which did it.
Let's rewind a bit here. One of the things that really characterizes English history is the near constant strained relationship between the Monarchs and the lower nobility, and later, the burghers. The Magna Carta is, of course, the most famous example, but really it had little impact on the long-term governance structure of the Kingdom. Instead, I'd argue that there are three events that represent the true origins of the 'Westminister' democracy.
First, we have the English Civil War. This represents the culmination of the simmering conflict between the parliament and the monarchy. However, 'parliament' in this case wasn't a body representing the people, but mostly composed of middle to lower nobility and a few burghers. Thus, it has more in common with the Baron's Revolt that led to the Magna Carta than a democratic revolution. At any rate, the beheading of Charles I, almost a century and a half before the French lopped of Louis XVII's head, is a microcosm of the event as a whole. In the French case, it was the supremacy of 'the people' in the democratic sense, while the English case was the supremacy of the lords of the land.
The second event was the Glorious Revolution. After the restoration of the monarchy in England after the Civil War, Charles II brought a period of calm to a degree; basically, he didn't do much to offend those who might oppose him among the lords. His successor, James II, could not say as much. He was a Catholic, which was bad, but worse, he tried to eliminate opposition, first through appointment, then by arrest, and particularly, pack parliament with his supporters. So, in response, parliament basically invited a new King in the form of William. Not only did they invite him and basically orchestrate the deposing of James II, they set the terms by which James (and his wife, the Queen regnant Mary II) were to assume and hold the crown. In essence, by this event they had assumed authority over determining who was a lawful monarch in the country. No more divine right of kings and all that jazz.
Finally, we have the events of the 19th century. This is a bit of a cop out I know, but bear with me here. During the 18th century, Parliament consolidated it's power. And again, 'Parliament' here mostly refers to the landed nobility and a few burghers. The 19th century changed a lot. In this case, Parliament was mostly motivated by the events outside of the United Kingdom. Starting in the late 18th century, you have the French Revolution and popular revolts elsewhere, in the mid 19th century, you have a series of republican and nationalist revolts across Europe, and almost everywhere, absolutism was being challenged and new liberal thoughts brought forward. Meanwhile, in England, a demographic shift meant that many of the electoral boroughs were a century or more out of date; some of them only had populations in the dozens or less, making them easy to control for anyone with money. The cities were woefully underrepresented in number in parliament, and this helped contribute to a growing call for reform. I'm not going to go over every reform act, but put simply, the industrialization and urbanization over the late 18th and into the 19th and 20th centuries helped seriously weaken that landed nobility's hold on wealth, and eventually, power. Some would argue they were motivated by revolts on the continent and in Ireland, others simply by the self-interest of industrialists who could use the new votes to their own advantage, in either case, the nobility could simply no longer muster the strength or the will to virtually monopolize the power in government, and thus parilament slowly reformed. First, by adding more seats for cities and giving more men the vote, and then slowly expanding the electorate and more fairly distributing the electoral boundaries, until we have the system we have today.
It is important to distinguish the decline of absolute monarchy from the establishment of democracy. Absolute monarchy was a contested idea for much of English history but the alternative was not democracy but greater power for the nobility and later for other elite groups. Although absolutism was finished by the end of the seventeenth century democracy was resisted, often violently by the state well into the nineteenth century and was arguably only fully realised with women's suffrage in the twentieth.
One of the interesting things about England/Britain is although the state was very powerful and centralised, the monarchy constantly had to compromise its authority to maintain its income, militarily and political stability so it was never as absolute as it pretended. The Whig historians of the nineteenth century tried to turn this mess into a grand journey towards the ideal system but it was never that clear.
One factor in the rise of democracy was the realisation in the later nineteenth century that the masses could be convinced to support established parties which led to a series of extensions of franchise by the Liberals and Conservatives who both expanded their appeal to these new voters. Many politicians of that era played the game superbly while having no ideological commitment to democracy.