The English Parliament started as the Great Council of William the Conqueror, and would gradually evolve from a council occasionally called to be the primary power in Britain. The first event I would mention is the signing of the Magna Carta, which gave the English people some basic rights and for the first time, put limits on the power of the crown. After the signing of the Magna Carta, many different kings tried to bully and control the English Parliament who at times lost and at times won. A major turning point in the history was the Provisions of Oxford, which forced the King Henry the 3rd to accept a new form of government. This government was ruled by a council of 24 members, 12 appointed by the nobles, and 12 by the crown. Although this was quickly forgotten as its inventor Simon de Montfort left on a crusade. He came back, but the agreement was scrapped and Montfort was killed. Despite not existing, the Provisions of Oxford are a key step, as afterward Henry the 3rd would regularly call a Parliament to appease his underlings.
Parliament grew as an institution heavily during the time of Edward the 1st. Edward wanted to unite Wales, Scotland and England under his rule by force. To achieve this goal he encouraged the submitting of petitions to Parliament to solve problems and have grievances heard. This established a precedent of Parliament responding directly to the people.
Parliament consolidated a huge amount of power at the end of Edward the 2nd's reign. Edward was very unpopular, and in 1326 Parliament met and decided to depose the King ( there had been no procedure before, so Parliament and through it the nobles decided it was Parliaments power). Thus Edward the 2nd was deposed in favor of Edward the 3rd.
Under the reign of Edward the 3rd, Parliament gained formal powers, in that it was established that no law could be made, nor tax levied without the consent of parliament. This came about because Edward was involved in the Hundred Years War and needed funds, which Parliament blocked until Edward agreed to the restraints.
The Tudors would mold Parliament into its modern two house structure, however they would consolidate the power of the King and Parliament would be weakened. After the Tudors died out, James the first of Scotland ascended to the English throne.
When Charles the 1st became King his handling of Parliament left a lot to be desired. They disagreed so often he dissolved them and ruled for 11 years without them; however, Parliament controlled the taxes, so to levy new ones, Charles had to call Parliament back into session. The two would squabble further until the English Civil War broke out. Parliament won this war and ruled without a King under Cromwell until 1659. The monarchy would be restored, but from this point on Parliament was no longer subservient to the King, they shared power.
When Charles II's son James came to power, there was a pretty large problem. James was Catholic and Parliament would not accept a Catholic King, so the Crown was taken from James and given to William and Mary. They were accepted as co-regents of England on the condition the English get the Bill of Rights. Thus the constitutional monarchy in England is born.
TLDR; centuries of struggle between the King and Parliament.
Questions, concerns? Feel free to comment. As always sorry if It is badly written, English is not my first language, but I tried.
StrategistEU has covered all the main points of how England (and Wales, and later Scotland) became a constitutional monarchy. Magna Carta, the split of Commons/Lords, the Civil War, the Glorious Revolution. However, there's a bit more that needs to be said about how it went from a Constitutional Monarchy to a Democracy, mostly in the 19th and early 20th centuries.
After the Glorious Revolution, the government of Britain (and later the UK) was essentially a Constitutional Monarchy. Now, the Monarch did have some powers (the power to declare war, choose a Prime Minister, and veto laws, for example), most of which (not all) the Queen today technically possesses still. However, the law-making and tax-raising powers of the King had been stripped away.
But this did not make Britain a democracy.
Firstly, the House of Lords was, as the name suggested, mostly made up of aristocrats, who had inherited their titles (there were also bishops of the Church of England, who had a much more active role in the House of Lords than they do now: most of the time, only 1 of the 26 Bishops actually turns up for votes today).
The House of Commons, by 1800, was comprised of MPs who were elected, but voters had to pass very high property tests, and after 1693, had to be male (not that many women could pass the property test, though a few widows could). Although it wasn't actually very common, a few infamous constituencies (known as "rotten boroughs") had only a handful of voters (by handful, this could mean as few as 7: rather unfair when other constituencies could have 10,000 voters) who sold their votes to the highest bidder. This, combined with the fact that voters were already only very rich people (about 200,000 in total, out of a population of millions), and the fact that MPs were not paid and so needed to be independently wealthy, meant that the House of Commons comprised of "Commoners" in legal status only. Your average MP was a wealthy member of the landed gentry, only slightly poorer than some of the aristocrats. In fact, many of them were from aristocratic families: they got elected by their father's friends and supporters in the constituency where he was the wealthiest lord, went to the House of Commons for 20 years, then moved over to the House of Lords when their father died and they inherited his title.
Alternatively, MPs could just be under the patronage (and pay) of wealthy aristocrats. Corruption was endemic and normalised.
One final thing to note is that since constituencies hadn't been redrawn in a very long time, a poorly populated countryside area might have 2 or 3 MPs, while a massive industrial city might only have 1 MP.
So, what we had in 1800 was government by the (rich, aristocratic) people, for the (rich, aristocratic) people.
By people, I of course mean men. Specifically, Anglican men: Catholics were barred from voting, and standing for political office, as were Jews, Anabaptists, Quakers, etc, etc. (Incidentally, the Whigs tried to let wealthy Catholics vote to keep the old Anglo-Irish elite happy when Ireland was incorporated into the UK, but the King vetoed it)
So, we come to the 1820s. Britain has recently defeated the horror (to aristocrats, that is) of revolutionary France, a nation created in a blood-soaked orgy of aristocratic head-chopping, not to mention non-aristocrats. But revolutionaries are found in other countries too. What if it happened in Britain? This was a serious worry for the British ruling class.
At the same time, with the Industrial Revolution, the middle classes were booming, and the Whig party wanted their financial support for the government.
Meanwhile, politically active members of the working class and middle class were agitating for the franchise to be widened: as you can imagine, something that rather worried their rulers, considering what had happened in France. Initially, MPs reacted to this by turning completely against any idea of electoral reform, but this later changed.
In 1819, radical groups began to hold huge rallies agitating for universal suffrage. Rallies in Birmingham (which had a total of 0 MPs, despite being one of Britain's largest cities) and Manchester attracted 10s of thousands of people. The government reacted with suppression, violence, and banning any meeting of over 50 people without permission.
But more and more MPs began to believe the franchise should be widened, either because they genuinely believed it, or because they thought it the best way to avoid revolution. When, in 1829, the Prime Minister, the Duke of Wellington (yes, it was still aristocrats in charge), gave Catholic land-owners the right to vote, anti-Catholic MPs began to believe that giving more power to the new industrial cities was a way to avoid Catholic influence. These cities were bastions of Non-Conformist Protestants (Baptists, Quakers, etc). Most of the British elite saw these Non-Conformists as rather vulgar and radical, but they preferred them to Catholics.
In 1830, there was a general election, in which electoral reform became a hot topic, with many middle class reform societies getting wide public support through peaceful rallies, pamphlets, etc. Wellington won again, just about, but after giving an anti-reform speech, had to resign 2 weeks later. Earl Grey became the new Prime Minister. He promptly tried to pass electoral reform but it was voted down. This led to widespread riots across the country, with several Bishops' palaces and Lords' manors being burnt or attacked. There was another failed bill, more rioting, and eventually the Tories abstained and let the Reform Act pass.
So now, everyone could vote, in fair elections with constituencies based on a recent census... by everyone, I of course mean only men in houses worth at least £10 a year, so only about 650,000. A significant number, but still very much a minority of the rich.
However, political agitation continued through the 19th century, and reform continued to be a hot topic. In 1866, the Liberals' (formerly known as the Whigs) leader, Lord Russell, tried widening the franchise slightly to "respectable working men" (ie the middle class) but the arguments that ensued within his party caused the government to collapse. But this brought the issue back into the spotlight. The Conservatives, who'd been out of power for decades, saw their chance, forming a government and introducing their own Reform Act. With pressure from mass demonstrations, this Reform Act ended up much more radical than the previous one, but the Conservatives believed the new voters would reward them with votes. So, the franchise was widened to the heads of most households, about 2 million men (out of 7 million). (This is also a lot to do with the rivalry between the two giants of late 19th century politics: Disraeli and Gladstone. But that's too involved to go into here)
At this point, with even some fairly poor men able to vote, further reform was fairly inevitable, especially with continued agitation by the Reform society, radicals, socialists, etc. In 1872 the secret ballot was introduced, which meant peasants were no longer intimidated into voting for their Lords' favoured candidate, in 1884 the franchise was widened further, and in 1918 all men over 21, and women over 30, were given the vote. This was equalised in 1928.
And that is how Britain became a democracy. Political campaigning, parties jockeying for votes, agitation, personal grudges, and occasionally a jolly good riot.