I would argue there are three major events that led to the modern political system, but of course also many minor events, gradual developments, etc.

​

Part 1

The first is the signing of the Magna Carta. Although this is often exaggerated in importance and effect, it is still important in terms of principle.
Early Norman Kings often took the quite sensible measure of asking advice from their council, which was generally made up of the most powerful lords both secular (Earls, Barons, etc) and ecclesiastic (Archbishops, Bishops, etc). This could even be the Magnum Concilium (Great Council). After all, if the King passes a law that all his leading subjects hate, how will he enforce it? He had no national army or police force, only the feudal oaths of his lords to supply men (or, as the system developed, cash payments). If all his lords refused to follow his laws, then he'd have no soldiers. So while there was not really a codified legal obligation for the King to consider the wishes of his lords when making laws, in practice it was a very good idea.

In 1100, the 4th son of William the Conquerer, Henry I, became King. The Barons didn't like him, the Archbishop didn't like him, his brother the previous King (who once threw Henry in prison for 2 years) had died in a hunting "accident" during a hunt that Henry attended. The common people, being Anglo-Saxons, didn't like him. He solved the last problem by marrying Edith, princess of Scotland and more importantly an Anglo-Saxon princess, great-niece of the last English King, Edward the Confessor. However, this made him even more unpopular with the Norman nobles.

So, to mollify them, he passed the Charter of Liberties. This charter bound the King to certain legal responsibilities; he couldn't seize property from a dead Bishop, he couldn't charge the sons of nobles to inherit their father's titles, he couldn't stop the widows or daughters of nobles from marrying, he forgave all murders and debts to the Crown from his brother's reign, he restored some laws of King Edward and William his brother had changed, he promised to maintain England's forests, he couldn't charge nobles tax and demand their soldiers, and he would no longer allow noblemen to pay money to escape prosecution for crimes.

These may not seem very related to modern politics, but it established a key principle; the King's power is not absolute, and he himself has obligations to the country, to God (ie the Church), to his nobles, and to fair and just laws. But... it was pretty much ignored for the next 115 years.

This brings us to the reign of King John. John was the younger brother of the King Richard I. He'd attempted unsuccessfully to rebel against Richard while Richard was on Crusade. But when Richard died, he won a war of succession against his young nephew Arthur. He inherited England, Ireland, Normandy, and extensive lands in France. But 3 years into his reign, in 1202, war broke out with the King of France when John abandoned his wife to marry another woman. Her fiancee complained to the King of France, and war broke out when John refused to go and answer to the King of France. He quickly captured the rebels, but kept them in bad conditions, and many died. John apparently killed his nephew Arthur while drunk and threw his body in the river with a corpse tied to it. He also treated his main ally very arrogantly. John's allies began to abandon him, and he hired many mercenaries. All of this made John more and more unpopular, and desertions in his forces to the King of France's side led to him losing most of his land in France.

John had gone from a powerful and rich King to a poor and unpopular one. He spent the next 10 years trying to rectify this by building up his forces and treasury through extensive taxes. Taxes for inheritance, taxes for imports and exports, taxes on Jews, taxes on widows, taxes on feudal lords to pay for wars even when there weren't actually any wars... you'll notice that some of these explicitly contradict the Charter of Liberty, something that, after 100 years of being ignored, the Barons suddenly remembered as being very important.
(A small note; John also did a lot of good work on currency standards, professionalisation of the courts, fair laws for commoners, etc. That's just not very relevant to this answer.)
Unrelated to this, John got into an argument with the Pope about the election of a new Archbishop, and the Pope excommunicated him. He used that as an excuse to raise money from the Church; seizing the lands of clerics who supported the pope, kidnapping their girlfriends and demanding fines to get them back, collecting tithes meant for the Church and keeping them, etc.

In 1214, John lost a war to France, again, and returned to England. With the King poor and weak, many barons revolted, swearing oaths to fight for "the Church and the Realm." They demanded John reissure the Charter of Liberties.

In 1215, John met the Barons to negotiate. He was forced to sign the Magna Carta, which promised many legal protections for freemen, guarenteed the freedom of the Church, and heavily regulated the King's rights to impose taxes without the consent of the Barons.

Most clauses have since been repealed or altered due to new laws, but three clauses still exist in English law, and some clauses have also been used in US trials.

• I. FIRST, We have granted to God, and by this our present Charter have confirmed, for Us and our Heirs for ever, that the Church of England shall be free, and shall have all her whole Rights and Liberties inviolable. We have granted also, and given to all the Freemen of our Realm, for Us and our Heirs for ever, these Liberties under-written, to have and to hold to them and their Heirs, of Us and our Heirs for ever.
• IX. THE City of London shall have all the old Liberties and Customs which it hath been used to have. Moreover We will and grant, that all other Cities, Boroughs, Towns, and the Barons of the Five Ports, as with all other Ports, shall have all their Liberties and free Customs.
• XXIX. NO Freeman shall be taken or imprisoned, or be disseised of his Freehold, or Liberties, or free Customs, or be outlawed, or exiled, or any other wise destroyed; nor will We not pass upon him, nor condemn him, but by lawful judgment of his Peers, or by the Law of the land. We will sell to no man, we will not deny or defer to any man either Justice or Right

The most significant clause was Clause 61, which established a council of 25 Barons, who were required to monitor the King and ensure he was following the rules. If he did not, they were legally authorised to fine him or remove his property. So here we have a)laws the King must follow, and b)a parliament of laws with power to enforce this on the King, and which was required to pass new taxes.

The King and the Barons soon fell out, of course, and the charter was not enforced. But John died a year later, and his young son became King. As the King was a child, he was weak, and while he won the civil war, was forced to issue a new form of the Magna Carta, watered down from John's version

Legal arguments over whether or not Magna Carta bound the King or not, but over time, large portions of it became accepted by both King and Nobles, with both sides ignoring it or fighting for it's rights as required.

Its legal strength retreated under the Tudors, but by this time the concept that the King ought to hold regular parliaments, and get their support for any new taxes (ie to pay for a war) was widely accepted.