I don't really understand what happened with Robespierre. He's basically all-powerful, condemning people to the guillotine and crushing all opposition. On 8 Thermidor he gave a speech announcing there were more traitors. Suddenly, the National Convention announces that Robespierre is, himself, a traitor, he's arrested and guillotined, and his policy of terror is outlawed. It seemed like he had their full support until he gave his speech on 8 Thermidor. What am I missing? What actually happened to make Robespierre fall so suddenly and dramatically? Was it really because of one speech, or was there something else happening behind the scenes?
Great question! I'm so glad you asked, because this is really 'the question' that got me interested in the French Revolution in the first place! As with any complex historical event, the French Revolution is condesnsed down until it's simple enough to fit into, say, a one-semester course (or, the horror, a two-week lesson plan!) As such, we have to not only simplify causality and eliminate events, but we also tend to give one-line characteristics to the players. King Louis XVI? Impotent moron. Marie-Antoinette? Evil spendthrift. And Robespierre? Arch-villain dictator.
Simplification is necessary—obviously you can't fit every important detail into any one book, course, documentary &etc.—but this oversimplification of Robespierre is so pernicious that it's my favorite to discuss and combat!
So the critical foundation for tackling this question is to dispell the idea that you summarize perfectly at the beginning of your question:
He's basically all-powerful, condemning people to the guillotine and crushing all opposition.
This is exactly the kind of oversimplification that is taught, but really doesn't follow the facts. Robespierre, to put it bluntly, was never as all-powerful as he is portrayed. It's FAR more complicated than that. He was never a singular point of policy, and was never a singular tyrant (it can be argued that the Committee itself was a dictatorship, but no one man held power within). This isn't to downplay his importance at all—even at the time he was seen as a powerful leader, and his leadership was absolutely influential. But the idea that he directly wielded absolute authority is very far from the truth. There are many different examples, but I'll provide the most poignant example of Robespierre's surprising lack of influence, that also calls into question his unequivocably "bloodthirsty" nature:
In June 1794, at the height of the Terror and (arguably) the height of Robespierre's power, he tried to intervene to save the King's sister, Madame Elisabeth, from the guillotine. He lost. His more literally bloodthirsty enemies on the Commitee—in this case (as in many others) this meant Biullad Varenne & Collot d'Herbois—would not entertain the idea of exile, but pushed for her execution. In this, as in many similar specific struggles, Robespierre was so far from all-powerful that he did not have the direct authority to intervene in who went to the guillotine.
Of course that's a tiny fraction of what could be said on that subject, but hopefully that will suffice to demonstrate the larger point: Robespierre was merely a big player among a group of 12 big players (the Committee of Public Safety) among a nation full of allies, enemies, and undecided. While he had name recognition and was undeniably one of the prominent leaders during the period of the Terror, he should be seen as a powerful voice (perhaps the most powerful voice) among many.
[For more reading on the Committee of Public Safety and Robespierre's role, I highly recommend Twelve Who Ruled by R.R. Palmer. It's very engaging and written in a narrative style that drops you into the action circa 1793. It also does a great job of showing how the responsibilities of the nation were divied up among the 12, and therefore sheds light on why Robespierre was not 'all-powerful' even within the committee itself.]
It's a very long story which has engendered a lot of debate but in essence this is what happened.
Robespierre's only power was that of persuasion. Through hard work, incorruptibility and his magnetic speaking voice, he was the single most important man in the Convention and the Jacobin Club as well as being one of the members of the Committee of Public Safety that really did have power. His say-so was necessary for the mass proscription of the Hebertists and the Dantonists (and assorted others crudely lumped in with them) in the spring of 1794.
However, he always had enemies on both extremes. His fatal mistake was that speech to the Convention on 8 Thermidor, which rambled on forever denouncing yet more traitors without naming anyone until just about anyone until a few found their voice largely through fear and turned on him. He and his closest supporters were then proscribed and sent to prison, but they were released on the initiative of the Paris Commune and went to the Hotel de Ville (Town Hall) to try and raise Paris against the Convention. They failed but it was a close-run thing.
The fall of Robespierre did NOT actually end the Terror; the three days of 10-12 Thermidor were among the bloodiest of all as those who had taken over eliminated almost anyone linked with Robespierre's crew. Many of those who ousted him had even more blood on their hands than he did on his - notably Billaud-Varenne and Collot d'Herbois in the Committee of Public Safety and the former 'terrorists' like Charrier who had far exceeded their powers when 'en mission' to rebellious areas. They formed a temporary alliance with those who really did want to wind down the Terror in order to get rid of Robespierre but it only really ended when they were outflanked in their turn. And that's leaving out the White Terror of later years...