There are many conflicts at work in Shakespeare's Julius Caesar, but I'll focus on what I take to be the three main ones: the triumvirs vs. the conspirators, friendship vs. the needs of the state, and personal ambition vs. democratic governance.
The first conflict, the triumvirs vs. the conspirators, is the most obvious, as it takes the form of a civil war onstage, rather than as an abstract concept floating around in the ether. In the play, the triumvirs—Octavius, Antony, and Lepidus—engage the conspirators—most notably Cassius and Brutus—in open warfare on the battlefield. They do so to avenge Caesar's death and bring the assassins to justice (and also, one can assume, to take control of the power vacuum left in the wake of Caesar's absence).
The second conflict, friendship vs. the needs of the state, is a little more abstract, as it unfurls within the confines of Brutus' psyche. Brutus and Caesar are known to have a strong relationship as close friends; however, Brutus has also come to fear Caesar's unchecked ambition and the potentially autocratic implications it could have for the future of Rome. As such, Brutus must decide whether his friendship with Caesar is more or less important than the health of the Roman state. In the end, Rome wins.
The last conflict, personal ambition vs. democratic governance, can be seen in many forms. First and most obviously, we can see it manifest in Caesar's desire to rule Rome singlehandedly and the conspirators' desire to stop him at all costs. Second, we can also see it in Antony's willingness to employ an angry mob to catapult him to a higher position of power, disregarding the fact that the conspirators were fighting to protect Rome's representative government. This conflict is the most far-reaching of the play's tensions, and its tragic outcome (personal ambition seems to defeat democracy) still resonates today.
There are three main conflicts in this play.
First, there is the conflict of friendship. Brutus and Caesar were friends. It was actually more than this. Caesar was like a father to Brutus. So, for Brutus to betray Caesar, there was a huge internal conflict. This is why if there is a tragic hero in the play, it is Brutus.
Similarly there was a conflict of national interest. Brutus was torn in his loyalties, because he strongly believed in and valued the Roman republic and he loved Caesar. In the end, he choose to defend the Republic (wrongly). There is an important historical detail to keep in mind, because when the republic was established, it was Brutus's ancestor (another Brutus) who started the consulship and the republic. So, his family line was steeped in republicanism.
Finally, there is a conflict between religion and reason. Right from the beginning of the play there were religious omens. For example, the soothsayer repeatedly said "Beware of the ides of March." There was also the storm and the dream of Caesar's wife. Caesar did not heed any of these things.
This play is primarily a man vs. self conflict. The protagonist, Brutus, must struggle with himself over his loyalty to his friend and his loyalty to his country. He must decide for himself if murder is justified in the name of preserving democracy. And, when the murder is committed, he must struggle with his guilt and his doubt of the loyalty of his conspirators.
This is also a man vs. society conflict. Brutus is struggling against the movement of his society away from the republic and towards a dictatorship. The people seem ready to accept such leadership, which is the cause of his actions.