What are the two sides of "the war" (internal conflict) going on in Brutus' mind in Shakespeare's Julius Caesar?
On the one hand, Brutus loves Caesar. This is a fact. The play brings this point out nicely, as Brutus is very reluctant to join the conspiracy. This is why he is the tragic figure in the play.
From a historical point of view, Brutus's loyalty to Caesar makes sense, because when civil war broke out in 49 B.C. between Pompey and Caesar, Brutus sided with Pompey. After the battle of Pharsalus, Caesar showed incredible clemency. He pardoned Brutus and many others. Moreover, Caesar accepted Brutus into his inner coterie of friends. He even honored him by making him the governor over Gaul. And in 45 B.C. Caesar granted him another honor by making him into the urban praetor. Hence, Caesar showed clemency, generosity, and honor.
On the other hand, Brutus is convinced by Cassius that Caesar wants to be made king. For example, Caesar is very popular with the people, and he has taken the title of dictator in perpetuity, which has never happened before in the history of the Roman Republic. Hence, Brutus is torn. He loves the Republic, but he also loves Caesar. I should also mention that Brutus's ancestor drove out the last king of Rome - Tarquin the Proud. Hence, Brutus's blood is Republican through and through.
In the end, Brutus sides with Cassius for the sake of the Republic, but he does so with a torn heart. Tragic, indeed.