Anti-Hero: Young Joe is a Nominal Hero at the start of the movie. He's (mostly) self-centered, has no qualms executing people for the mob, and shows few signs of heroism throughout most of the film. He develops more into an Unscrupulous Hero when Cid becomes something of a Morality Pet for him in the third act.
Byronic Hero: Joe himself is a hitman who not only works for himself, but is also a flawed protagonist with a mysterious past that is never mentioned.
Functional Addict: Averted. While relatively efficient at their jobs, both Joes are crippling addicts to the "eyedrop drug." Somewhat played straight with Old Joe however, as he's been off the stuff thanks to his wife. The withdrawal symptoms on the other hand look to be similar to heroin, only about a hundred times worse.
Heroic Suicide: Kills himself to stop the cycle that would create the Rainmaker.
Taking You with Me: Kills himself in order to erase Old Joe from existence and (hopefully) stop Cid from becoming the Rainmaker.
Took a Level in Kindness: Young Joe is initially very greedy and selfish, prefering to rat out a friend rather than losing half of his silver, but he became a better person after his character development and by the end of the film he even sacrifices himself in order to save the lives of Sara and Cid.
Always Save the Girl: Old Joe will do just about anything to save his wife. However, when given an option to stop his past self from meeting her, he refuses. He cannot sacrifice their relationship.
Anti-Villain: Has sympathetic goals when trying to save his wife and prevent the rise of a merciless underworld kingpin. However, he's unwilling to sacrifice his relationship with his wife to save her, and is willing to kill children to stop the kingpin (though he's also tormented by guilt when he does).
Determinator: He won't stop at nothing, not even at killing children, in order to save his wife in the future.
Evil Old Folks: Exceptionally good at combat despite being in his 50s, which puts him at least a decade older than just about everyone he goes up against.
Future Badass: While Young Joe, as a Looper, is a killer and criminal, his weapons experience basically boils down to shooting bound and gagged prisoners with a blunderbuss. Old Joe, however, has a few decades of hardened crime life under his belt, and is therefore much more of a threat.
He Who Fights Monsters: By trying to stop the events of the past to save his wife, he becomes a child-murdering monster.
Hypocrite: He accuses his younger self of being selfish, but he himself is not willing either to give up on his love to save her life in the future.
Action Mom: Handy with a shotgun when it comes down to it.
Badass Boast: Her first major line has her delivering one while aiming the rifle at the apparent intruder in the corn field. Played with in that Joe quickly realises she's overcompensating - as she's spending too much time describing what she's going to do than actually doing it.
Mama Bear: Sara makes an effort to be this trope personified, except she's way softer than she plays at. Most of her badass is bluster, which Joe twigs to quickly.
Sara: This is a Remington 870. One blast could cut you the fuck in ha— Joe: (cutting her off) In half. Yeah, that's telling. You're holding a gun, I say I'm not afraid, so you describe the gun to me. It's not the gun I'm not afraid of.
Mind over Matter: She's got genetic telekinesis powers, or TK, and they're a bit stronger than most people's. In her wild youth, she'd challenge would-be suitors to move a quarter, then hold it with her own power for her own amusement. Her son got them with compound interest.
Morality Chain: Cid's transformation into the Rain Maker is implied to have been started by watching her get killed. Her influence is what causes him to spare older Joe.
Reluctant Warrior: Downplayed. Sara doesn't want to kill anyone, but has to protect her farm from vagrants in some way. She uses a shotgun loaded with rock salt; highly unlikely to be fatal, but it hurts like hell.
Decoy Protagonist: Well, Decoy Deuteragonist. He's set up to be important in the first act, but stops being relevant to the plot shortly after becoming important to it.
Fate Worse than Death: Implied for young Seth. Abe knows they can't kill him (because he needs to survive long enough to be sent back in time and complete the loop), so instead condemns him to 30 years bedridden with no limbs, tongue, ears or nose. The fact he can (presumably) still talk at least lets him narrowly avert And I Must Scream.
Played By: Frank Brennan
And I Must Scream: Averted, Seth begins to scream in horror as he watches his limbs and fingers disappear, one by one, only for his tongue to disappear as well, making him unable to speak properly but is cut short with a bullet to the head.
Avenging the Villain: His entire involvement in the final showdown is motivated by his desire to avenge Abe's death at Old Joe's hand.
Big Bad Wannabe: Blue has aspirations to be something greater, but he's far too incompetent to be taken seriously.
The Chew Toy: Things rarely go right for Kid Blue, although it's usually his own fault.
Determinator: Credit where credit's due; he doesn't give up, even if it's the smart thing to do. Especially since this involved driving a hoverbike with a single broken hand, pulling off reasonably accurate shots with his offhand in the meantime.
Chekhov's Gunman: Two examples. Her son turns out to be one of the potential candidates to become the future Rain Maker. Old Joe visiting her apartment also leads to Abe's crew catching him.
Don't You Dare Pity Me!: Implied. She turns down Joe's offer to give her half his silver, insisting she'd rather earn her money on her own.
Hooker with a Heart of Gold: She seems good-natured enough, and comforts Joe when he's feeling guilt over Seth's death. She's also working as a prostitute to provide for her son. However, she flatly rejects Joe's attempts to "save" her from her lifestyle, and she sees him as just another client.
Break Them by Talking: His preferred method of dealing with troublemakers. A few minutes' conversation, and Joe is ready to give up Seth's location rather than lose half his silver. If talking doesn't work...well, that's what the hammer on Abe's desk is for.
Ambiguous Disorder: There's something a little odd about Cid. He's very mature and intelligent for his age in some ways, while also susceptible to age-inappropriate tantrums and meltdowns which, combined with his TK abilities, give him the ability to literally tear people he doesn't like to shreds. He also has a need for control over his environment, which suggests he may be somewhere on the autism spectrum.
Ambiguous Start of Darkness: Shows some creepy and amoral tendencies before Old Joe goes after his mother, making it unclear if Young Joe saving his mother will ultimately stop the Rainmaker.
Roaring Rampage of Revenge: In the timeline where he becomes the Rainmaker. He watches his mother be killed, and that becomes his Start of Darkness. He eventually orders all Loopers killed, presumably trying to prevent his mother's death.