Why is the mob important in Julius Caesar?How do they fluctuate from one extreme to another?

The mob is important in Julius Caesar because of their power. Anyone who can command the allegiance of the mob can control the city of Rome itself, and anyone controlling the city of Rome is more than half-way towards total control of the Roman empire.

All of the major players in Julius Caesar recognize the power of the mob. However, some of them understand what moves the people more accurately than others, and are so rewarded with greater success.

The conspirators are uneasily aware of Caesar's popularity with the common people as they are plotting his assassination, and it is probably no coincidence that Casca, who gives a half-contemputous account of how well the people love Caesar (Act I, Scene 2), is also the first to urge Brutus to take the podium to justify Caesar's murder (Act III, Scene 1). A bit further on in the same scene, Brutus says to Mark Antony,

Only be patient till we have appeased
The multitude, beside themselves with fear,

Thus, the conspirators understand the power of the mob and the need to get the common people on their side. However, Brutus goes about this in an intellectual rather than an emotional way, and is so swept away by notions of fairness and justice that he allow Mark Antony to speak at Caesar's funeral service. Cassius is horrified by this, realizing what Antony might be able to do by appealing to the mob:

You know not what you do. Do not consent
That Antony speak in his funeral.
Know you how much the people may be moved
By that which he will utter? (Act III, Scene 1)

Brutus ignores this advice, with the result that his own justifications are forgotten by the popular masses when they hear Antony's passionate appeal to their sentiments, and the conspirators are run out of town (Act III, Scene 2) by a lynch mob so furious that it even murders one man for merely having the same name as one of the conspirators (Act III, Scene 3).

Thus, the mob is important in Julius Caesar because they hold the key to power in the city of Rome, capital of the Roman Empire. The conspirators' failure to win them over highlights their lack of understanding of the political and military situation, and foreshadows their defeat and death.