Compare and contrast the characters of Coriolanus and Aufidius as well as their relationship with each other.

Coriolanus and Aufidius begin the play as military leaders from opposing sides. While Coriolanus is a Roman general, Aufidius leads the Volscians. Coriolanus is unyielding, brutal, and relentless in battle; similarly, the same can be said for Aufidius. If there's anything the two have in common at the beginning of the play, it's their prevailing and consuming hatred for each other. Each man's chief goal is to annihilate the other in hand-to-hand combat.
Despite their mutual hatred, however, the two men clearly admire each other. This can be seen in their meeting in Act 4 Scene 5, when Aufidius generously calls his arch enemy "noble Marcius," "worthy Martius," and "Mars" (the god of war). In fact, Aufidius really lays on the compliments, at one point using a sexual analogy and obvious sexual innuendoes to characterize his image of Coriolanus. Indeed, some of Aufidius' words may have made the typical English theatergoer blush:

Let me twine Mine arms about that body...Know thou first, I loved the maid I married; never man Sigh'd truer breath; but that I see thee here, Thou noble thing! more dances my rapt heart Than when I first my wedded mistress saw Bestride my threshold.
...thou hast beat me out Twelve several times, and I have nightly since Dreamt of encounters 'twixt thyself and me; We have been down together in my sleep, Unbuckling helms, fisting each other's throat, And waked half dead with nothing (from Act 4 Scene 5).

Aufidius maintains that Coriolanus looks better standing before him in his unkempt, savage state than his wife did on the day he (Aufidius) married her. Not only that, Aufidius confesses that he dreams of tackling Coriolanus in combat, where they undress each other in the heat of battle—dreams from which Aufidius wakes up "half dead with nothing."
Earlier in the play, Coriolanus admits that he thinks of Aufidius as a "lion" that he's "proud to hunt." Both warriors openly express their admiration for each other in their meeting. Coriolanus maintains that, had he been a man to fear death, he would have "of all the men i' the world" avoided Aufidius. However, he only comes to Aufidius "in mere spite, To be full quit of those... banishers" who have sentenced him to exile from his Roman homeland. He tells Aufidius that he means to join with him and to fight against his "canker'd country with the spleen of all the under fiends." For his part, Aufidius welcomes his adversary warmly. He intends to use Coriolanus' skills and knowledge (of Rome's strengths and weaknesses) to his advantage.
Here, we can see that Aufidius is more of a tactician than Coriolanus. He knows that Coriolanus has put himself in his grasp, but he doesn't take advantage of his enemy immediately. Instead, Aufidius concentrates on his goal to defeat and subjugate Rome; so, he sees beyond the moment and chooses not to focus on vengeance. On the other hand, Coriolanus is intent only on avenging the loss of his station and power. He fails to see that his alliance with Aufidius may prove to be a double-edged sword. Coriolanus is too focused on the surface, immediate possibilities of the partnership; in the heat of his emotional angst, he has failed to grasp the logistics of such an alliance. This is a major difference between both men.
In the end, at the behest of Volumnia and Virgilia (his mother and wife respectively), Coriolanus withdraws from battle with Rome. Aufidius is none too pleased with his ally's decision and characterizes his lack of fortitude as an act of cowardice. He brands him a "traitor," someone who breaks "his oath and resolution like a twist of rotten silk." Meanwhile, Coriolanus accuses Aufidius of being a liar. The argument between both soldiers end in Coriolanus' death at the hands of Aufidius' men.
Aufidius is ever the master tactician; he uses situations to his advantage. Even after he is betrayed by Coriolanus, he resorts to pointing out Coriolanus' sins before the people (how Coriolanus has "widow'd and unchilded many a one" in Rome) to argue his case. Aufidius cleverly lays the foundation for his final action: the act of presiding over Coriolanus' death. By hook or by crook, Aufidius aims to prevail, and he does. For his part, Coriolanus has definitely underestimated his arch enemy!