Foil
Character Role Analysis
Ferdinand and the Cardinal
Although they're teamed up against the Duchess and her family, the Aragonian Brethren (in another life, Ferdinand and Cardinal were clearly in a motorcycle gang) are distinctly different men, and frequently contrasted against each other: Ferdinand's hot-headed and passionate (and, eventually, crazy) while the Cardinal is calm and calculating.
Think of the scene in Act 2 where they find out that the Duchess has given birth: Ferdinand totally loses it, raging about her "infected blood" and "cunning bawds." The Cardinal? Cool as a cucumber:
You fly beyond reason
[…]
I can be angry
Without this rupture; there is not in nature
A thing that makes man so deformed, so beastly,
As doth intemperate anger. (2.5.46, 55-58)
By the end of the play, both brothers end up as victims to the extremes of their personalities: Ferdinand goes nuts and becomes convinced he's a werewolf (literally believing himself to have transformed into the "deformed, beastly" thing the Cardinal warns him about) while the Cardinal ends up on the wrong end of his own plots. His own assassin kills him while his courtiers stand idly by, per the Cardinal's own orders.
Antonio and Bosola
Although these two are different in almost every way, they share something important: they're both commoners working for aristocrats in hope of advancement. Antonio goes the honest route: he's worked for the Duchess for a long time and has risen to steward, the highest position you can get in the court if you're not an aristocrat. In a world where everybody's scrabbling for promotion, Antonio professes to be happy where he is, telling the Duchess, "Ambition, madam, is a great man's madness" (1.1.412).
Bosola, on the other hand, figures that normal service is a sucker's game. As with Antonio, it's not clear that Bosola's after social advancement, and his reasons for becoming a spy for Ferdinand are complicated (go check out his "Character Analysis" for more). Although Bosola and Antonio work side by side in the Duchess's court, they're on opposite teams—Antonio works faithfully for the Duchess, whom he loves, and Bosola works equally faithfully for Ferdinand, whom he hates.
They both have, in their own ways, deep devotion to their service, both for very different reasons and with very different results. Bosola, too late in the game, discovers that Ferdinand isn't going to reward his service, and his transformation to avenger comes in large part becomes Bosola realizes that the Duchess is the prince he should have served, and that in her service alone "some preferment in the world can yet / arise from merit" (3.2.281-82).
The Duchess and Julia
Not an obvious pairing, but these two ladies are the only women you see exhibiting sexual behavior in a play that puts a lot of focus on beliefs about female sexuality. Julia is the wife of the aging courtier Castruchio, whom we're led to believe won't be landing the cover of GQ anytime soon. She's having an affair with the Cardinal, but when she sees Bosola she's immediately smitten.
She starts putting the moves on him big time, telling him,
[…] now I woo you.
[…]
We that are great women of pleasure, use to cut off
These uncertain wishes and unquiet longings
[…]
Had you been in th'street,
Under my chamber window, even there
I should have courted you. (5.2.177, 187-92)
This should remind you strongly of when the Duchess proposes to Antonio, where she declares "The misery of us that are born great, / We are forced to woo because none dare woo us" (1.1.433-34).
Between Julia and the Duchess, there's a lot of wooing going on, but they're written as being on opposite ends of the spectrum: the Duchess is a single woman who is "born great" and is proposing a legitimate marriage to a genuinely decent guy she's been into for a while. Julia is a "great woman of pleasure" who's cheating on the guy she's cheating on her husband with so she can hit on an assassin she met three minutes ago. Stay classy, Julia.
Even though people are accusing the Duchess left and right of having committed a social and sexual crime by marrying Antonio, next to Julia it definitely looks like she's on the straight and narrow.