Some people think we don't have a constitution. But we do. What we don't have is a single document entitled "The Constitution". The British Constitution is contained in a series of documents, which, taken together, are our constitution. Some people in the US think they have a single document entitled "The Constitution", but they too have a series of documents which, taken together, comprise "The Constitution"
The best known of our documents, is Magna Carta, the Great Charter, of 1215. Actually, that was annulled in 1216, but was subsequently changed and re-issued, changed again and re-issued, and it doesn't say what many people think it says. Nevertheless, it's an important early document.
Another important document is the Bill of Rights, 1689, limiting the power of the monarch and setting out the rights of parliament.
At the head of the British political system is the monarch, currently Queen Elizabeth II. She is also the head of the Church of England. She has no power whatsoever. She launches ships, opens supermarkets and parliament, and is an immense tourist attraction.
The real sovereign power in the UK is the House of Commons, also known as the Mother of Parliaments, because so many other countries have copied our system. There are 650 seats (that number can change with changes in electoral boundaries), each elected by a constituency using a first-past-the-post system. That system favours the two large parties, punishes any small parties and is a huge deterrent to either of the two big parties to split. There is also the House of Lords (also known as the Upper House), which is unelected (currently 800 members). Lords (and Ladies) are "appointed" and not elected. There is constant agitation for this to change to an elected Upper House, but there's no sign of this happening soon. The HoL has 92 hereditary peers (this is a fossil of the 1066 Norman Conquest), and 26 Lords Spiritual. The Lords Spiritual are archbishops and bishops of the Church of England, and is the only impact of religion on British politics, because whenever any politician expresses religious faith, they are subjected to prolonged and derisive laughter, unlike the situation in the US, where loudly and repeatedly expressed religious belief appears to be a necessity.
A few definitions - there is a kingdom called England, a kingdom called Scotland (but the UK monarch combines those roles), a principality called Wales (the monarch's eldest son is the Prince of Wales, and there's Northern Ireland, also called "the six counties" or "Ulster". The UK is all of the above. "Great Britain" is England, Wales and Scotland, but sometimes people use it to mean the UK. The term "Britain" is very poorly defined, it could mean just the largest island, or the whole political entity. We have a thing called "devolution" which means that Scotland, Wales and Northern Ireland all have their own parliaments, with various amounts of power. The Scottish parliament (also called "Holyrood" after the location where it sits) has a lot of power; they have, for example, enacted that university education should be free. The Northern Ireland parliament (also called "Stormont" after the location where it sits) has some power too. The Welsh Parliament (actually called the National Assembly for Wales) has very little power.
When the UK joined the EU, we agreed that our HoC would pass any regulations that the EU passed. Actually, we joined the EEC (European Economic Community, we had a referendum in 1975), which was very different. First of all there were only nine members, and secondly is was more about free trade. The EU (European Union) has 28 members, and is committed to "ever closer union", meaning we're trying to become a kind of "United States of Europe".
But there's always been people in the UK who weren't too keen on the EU (known as Eurosceptics) of which Nigel Farage (who loves to pose outside a pub holding a pint and a fag pretending to be a man-of-the-people although he's actually a millionaire ex-stockbroker) was the loudest mouth. And in 2015, the current Prime Minister (PM) David Cameron, in order to fend off UKIP (the UK Independence Party, a one-issue party headed by Farage whose only objective was to leave the EU) and to avoid a split in his own party (there being many eurosceptics and many europhiles in the party), promised to hold a referendum on leaving the EU (British exit, or Brexit) on an assumption that he made that Brexit would be rejected. In June 2015 the referendum happened amid a welter of misinformation emitted by both Leave and Remain, and to everyone's surprise, we voted Brext, 52% against 48%, and since Cameron had neglected to set any threshold for the referendum, we were committed to leave. Cameron resigned, there wa a brisk and brutal fight amongst the grandees of the party, and Theresa May (formerly Home Secretary) emerged as the only candidate after all the others "voluntarily" dropped out.
Brexit is the worst idea we've had since 1169, when we invaded Ireland.