Most good stories start with a fundamental list of ingredients: the initial situation, conflict, complication, climax, suspense, denouement, and conclusion. Great writers sometimes shake up the recipe and add some spice.
Initial Situation
Caesar is a national hero, and there are rumblings in the Senate that he seems to be on the path to becoming a king.
Caesar has retuned to Rome after fighting and killing Pompey, his former co-leader in the Roman triumvirate. The irony of this is lost on the plebeians (common people), who celebrate the individual instead of the nation. The situation at the beginning of the play is tense, and it's clear something needs to happen to break the tension.
Conflict
Cassius is gathering forces to rebel against Caesar, which amounts to treason. Brutus must be convinced to join the plot.
Caesar's slow ascendance toward kingship and absolute power worries those who think the plebeians are ignoring what will be an inevitable tyranny under Caesar. Though Cassius has been trying to incite Brutus to rebellion by suggesting that Brutus is better than Caesar, Brutus ignores this and is moved by the fact that Rome must be greater than Caesar alone. Brutus receives a (fake) letter that confirms things in Rome are really pretty bad, and he's the only one that can do anything about it. His personal convictions about Caesar have to be overcome by his patriotic commitment to Rome.
Complication
The conspirators agree that nobody touches Antony, which unsettles Cassius. When Antony comes to see Caesar's body, he convinces a trusting Brutus to let him speak at Caesar's funeral.
The conspirators pull off the murder easily enough, but they're hurt by Brutus' naïve thinking that everyone will love them after they calmly explain their true motivations: that they only murdered Caesar for the good of Rome. The people are panicking, which was not part of the plan. Antony takes advantage of Brutus' trust by asking if he can give a speech at Caesar's funeral. Though Antony promises Brutus he'll lay no blame, he fully intends to have Caesar avenged by inciting the people to riot.
Climax
Antony gives a rousing speech; public opinion turns against Brutus and Cassius.
Antony uses masterful rhetoric to highlight the terrible dishonor of the traitors. On the surface, he claims the traitors are all honorable men, but he subtly undermines his claims by showing all the wrongs they have done to Caesar. This gives the people reason to be incensed on Caesar's behalf. They take to the streets, screaming "Burn! Kill!" etc., etc., which really tickles Antony, who knows his diabolical plan has worked.
Suspense
Brutus and Cassius have fled; Antony is joined by Octavius and Lepidus; Brutus and Cassius are fighting.
Brutus and Cassius have fled the city. In the meantime Antony is busy plotting with his two friends, Octavius and Lepidus, whom he will likely betray as well. Even our would-be heroes, Brutus and Cassius, quarrel over some shady business deals Cassius has made. Everything seems to have been for naught: Antony and his friends will become the new tyrants, Cassius seems to be showing his true colors as dirty and underhanded, and Brutus will have lost everything without succeeding in saving Rome from tyranny. What good can possibly come of this?
Denouement
Portia dies; Caesar's ghost shows up; Brutus and Cassius make a suicide pact.
Everyone comes to their senses when Brutus announces that Portia has died. That single change begins to shed light on Brutus' sense of the futility of this fight. Caesar's ghost hasn't helped the matter, but Brutus is brave as ever, saying he'll meet the ghost when it comes. Cassius has a poetic reckoning with himself too – it's his birthday, so if he dies now his life will have come full circle. The two friends make a subtle agreement to never go back to Rome in chains, which implicitly means they'll take death before dishonor. The audience can now rest easy that either way the men are nobly resigned to die with honor, even if it's by their own hand.
Conclusion
Cassius and Brutus kill themselves; Antony and Octavius (with Lepidus) are free to rule Rome.
For once, this is exactly how it all played out in history (which Shakespeare often has a funny habit of revising). Having made peace with each other and themselves, Brutus and Cassius kill themselves when their battle against Antony and Octavius seems to be lost. Antony and Octavius take the field where Brutus has fallen, and Brutus is proclaimed the noblest man in all of Rome. Unfortunately, he's a bit too dead to hear it, and that praise likely didn't matter to him anyway.