Titus Andronicus is a general and a war hero who has just returned to Rome after a ten-year battle against the Goths. Titus gets into a vicious cycle of revenge with Tamora (Queen of the Goths turned Roman Empress) that culminates in an act of cannibalism.
The thing to know about Titus is that he's pretty old-school, which means he's a stickler for tradition and custom, and he's super loyal to Rome. He's also a pretty lousy dad. We could argue that this is what gets Titus into trouble and causes so much of his (and his family's) suffering. Let's discuss.
In the play's first scene, we learn that Titus has spent the last ten years of his life fighting for Rome against the Goths, and he's lost 21 sons (!) to this nasty war. Titus is adamant that Tamora's eldest son, Alarbus, should be ritually sacrificed in order to appease the spirits of the dead.
Even after Tamora pleads for her son's life, Titus insists on carrying out this Roman custom: "Religiously they ask a sacrifice: / For this your son is mark'd, and die he must" (1.1.3). Hmm. Looks like Titus has just given Tamora a reason to come after him, don't you think? This act of ritual sacrifice is what paves the way for the bloody cycle of vengeance to come.
Titus also insists that Saturninus should be emperor because he's the "eldest son" of the late emperor of Rome (1.1.11). So, Titus is following the old-school rules despite the fact that Saturninus has just threatened Titus and hasn't given us any reason to think he'd be a good ruler. (Another huge mistake, given that Saturninus makes Titus's enemy, Tamora, his empress.)
And don't even think about standing between Titus and his beloved traditions. Remember what happens to Mutius? Titus doesn't think twice about killing his own son for interfering with Lavinia's engagement to the new Roman emperor (1.1.17).
Are we surprised at this point? A little, but Titus has already shown us that his loyalty and service to Rome matter more to him than the lives of his sons. In other words, Titus isn't such a great dad, which he later confirms when he kills his daughter Lavinia so her "shame" and his "sorrow" over her rape will die with her (5.3.5).
For a lot of literary critics (like Katharine Eisaman Maus) Titus Andronicus is a typical (or even "exaggerated") version of the 16th century revenge tragedy hero. He starts out as a law-abiding citizen, but then his quest to right some kind of injustice sets him on a destructive path that calls into question his own morality. (See the introduction to the Norton Shakespeare's Titus Andronicus for more on this.)
The "injustice," of course, is that Titus's loyalty and commitment to Rome are rewarded by Saturninus's ingratitude and the Empress's plot to devastate his family. On the one hand, you could argue that Titus's response is completely justified – when Titus kills Demetrius and Chiron and serves them in a pie to Tamora, the offenders get their "just desserts." On the other hand, Titus has given Tamora a very good reason to go after him.