The history of Margatet Thatcher's Prime Ministership is actually well laid out in this Wikipedia entry. I will therefore overlay some personal commentary on that entry for this part of the narrative.
In 1987, the Conservative Party once again easily won the General Election, against a Labour Party that was still unable to articulate a convincing competitive governance message. The party was still deeply fractured over political philosophy, with the hardline socialists still in the ascendancy. Advised by US political experts, the Labour leader Neil Kinnock put himself and his ministers into sharp suits (the previous Labour Party leader, Michael Foot, had attracted a lot of negative comment for his scruffy style of dress), and Labour Party rallies began to look like US party conventions, jarringly so. I remember watching a Labour Party event and thinking "this looks like American showbiz" (and not in a good way). The marketing failed to cover up the divisions in the party, and the very real lack of a coherent alternative to Thatcherism, and Labour was defeated once more.
Thatcher was now safe until (at the latest) 1992. There was a massive economic boom under way, which was making a lot of people happy, although large parts of the old industrial areas of the UK were suffering badly from precipitous decline.

From a personal political standpoint, Margaret Thatcher now entered a new and more dangerous period in her Prime Ministership, a period marked by increasing disputes with members of her own cabinet.
The first event that made people take notice was when Michael Heseltine, the Defense Secretary, abruptly resigned during a Cabinet meeting as the result of a dispute over a military helicopter deal with Westland. The cause of the resignation was a dispute between Thatcher and Heseltine over the future of the helicopter industry in the UK. Heseltine famously walked out of 10 Downing Street in the middle of the meeting to return on foot to his office down the road and round the corner, and breezily informed pursuing journalists "I have just resigned from the Cabinet". Heseltine was one of the higher profile and publicly valuable members of Thatcher's government. He was youthful, good looking, articulate and telegenic, a useful antidote to dull and more sinister looking members like Nicholas Ridley and Norman Tebbit. However, Heseltine did not need the income from a government position. His family owned Haymarket Publishing, one of the largest publishing firms in the UK, so for him, the government salary was merely pocket change. As a result, when he came face to face with what he considered a matter of principle, resigning did not leave him unable to pay the mortgage, and he took a stand and left the cabinet.
The fallout from what became known as the Westland Affair also cost Thatcher the resignation of another minister, Leon Brittan. To paraphrase an old saying, to lose one minister could be said to be unfortunate, to lose two begins to look like incompetence...

By 1987, Thatcher's policy focus seemed to have shifted to a change that would come to define her later years, and not in a good way - the introduction of the Community Charge (as it was officially known) or the Poll Tax as it was known to most people.
The Community Charge was intended to replace the rating system (which, for people in the USA, was and is the property taxation system in the UK). The rating system was vaguely progressive, it taxed people according to the value of their property. The Community Charge was not, it taxed individuals, so it penalized lower-income households, which tend to be larger than higher-income households. It was therefore deeply regressive compared to the rating system. In that respect, it clashed deeply with UK culture, which tends to be more egalitarian than the USA.
Thatcher insisted on trying to implement the Community Charge despite opposition from within her own ministers, Conservative Party advisors (who correctly determined that it was electorally unpopular) and the media. She had moved into the zone of implementing change, not because it made sense, but because ideology told her it was right.
The Community Charge dispute, coupled with the increasing tendency of Thatcher to listen to her new economic advisor Sir Alan Walters instead of her own ministers, led to the departure in 1989 of Nigel Lawson, the Chancellor of the Exchequer. Lawson had been Chancellor for a long time, in what is normally a fungible political position (Chancellors in the UK tend to get blamed for everything bad, including the weather, and the position wears down even the most resilient of people), so his resignation was a major political event.

By the summer of 1990, Margaret Thatcher was on a shortening rope in terms of power. She had alienated a number of people in her own party. At the same time, her domestic popularity was at an all-time low. A recession caused as a reaction to the mid 1980's economic boom had eroded her popularity and that of the Conservative party.
The final blow, which essentially led to her opponents gaining critical mass, was the resignation of Geoffrey Howe from her government in November 1990. Within days of his resignation, rumours began to spread that disaffected party members were working to force her to resign. Under the somewhat arcane rules of electoral politics in the UK, there would be a General Election due by the end of 1992, and the Conservative Party feared, that with Thatcher as their leader, they would be defeated.
One thing that politicians value above oxygen and publicity is power or the prospect of power. They therefore began to organize to replace her.
A true personal story; the weekend that the Conservative Party crisis came to a head, my wife and I flew to the Maldives for a vacation. We left on the Saturday, with rumors swirling of a vote of No Confidence being planned in the House of Commons. The general view was that Margaret Thatcher was doomed, and it was only a matter of time before she resigned.
On Tuesday morning, at breakfast on the island, our travel rep. brought us to order by banging on a water jug with a spoon. "I have some news for you", he shouted."Margaret Thatcher resigned last night".
The whole table of 20+ vacationers (and I mean the whole table) broke out in cheering and clapping. The Maldives is not a blue-collar destination. We had managers, lawyers, accountants, and IT people on this table, all white-collar people who had benefitted from the Thatcher governments. Yet, to a man (and woman) they were apparently glad to see the back of her. That single incident told me how badly she had overstayed her welcome and her time in office, and how she had been reduced to (as they saw it) a Pain In The Arse over time.