Finnick dair is one of District 4's beloved Hunger Games victors. Like Katniss, he's good fighter, but he basically won his Games because he's drop-dead gorgeous, not to mention telegenic and charismatic. We'll let Katniss explain:
Finnick never wanted for anything [in the arena], not food or medicine or weapons. [...] When he received a silver parachute with a trident – which may be the most expensive gift I've ever seen given in the arena – it was all over. (15.25)
Finnick had help from the outside and used it to his advantage. If there's anyone who could win the Quarter Quell against Katniss or the highly trained tributes from District 1, it's probably Finnick.
Finnick is a walking contradiction: one of the youngest and most innocent people ever to win the Games, he now seems to be trying to forget about it by throwing himself headlong into a life of extravagance. While Haymitch has lost himself in alcohol, Finnick has lost himself in high-style Capitol living. Handsome, sexy, and flirty, he seems entirely caught up in the pleasures of the flesh. He has a scandalous reputation in the Capitol as someone who knows – and sleeps with – everybody. When Katniss first meets him, he comes off as a total sleazeball.
Is Finnick really that skeezy, or is he simply playing for the cameras, just like Katniss? In the arena, he turns out to be a great ally. He transforms from a slimy guy attempting to seduce Katniss with sugar cubes to tough fighter and a good strategist who, surprisingly, always seems to have Katniss' back. Katniss says it herself:
By the time we pull ourselves together, I'm thinking that maybe Finnick Odair is all right. At least not as vain or self-important as I'd thought. Not so bad at all, really. (22.42)
Finnick also has hidden depths. He loved his mentor, Mags, like family, and had to watch while she sacrificed herself in the arena to save him, Katniss, and Peeta. What's more, Mags had already sacrificed herself just to be in the arena, by taking the place of Annie, the woman Finnick loves.
By the end of the Games, Finnick is seriously depressed. He's had to watch his friends and mentor die, and he's had to endure the emotional torture of imagining Annie in pain. When he and Katniss are pulled out of the arena and away from the Capitol, Annie is nowhere to be found. She's in danger back at home, just like Peeta is in danger in the Capitol.
In many ways, Finnick and Katniss are a lot alike. They both care deeply for a fellow victor and feel like they have to protect him or her. They both won their own Games handily, and they're both excellent fighters. If anything, Finnick is even more selfless than Katniss. He serves the revolution. He knew the story going into the arena and was ready to die in order to save Katniss' life.
(Dying to know more about Panem's favorite prettyboy? Read all about him in Mockingjay.)
Mags has a small but pivotal role as a tribute in the Quarter Quell. Like Finnick, she is a victor from District 4. From her first entrance Mags shows her grace and courage, as well as her self-sacrificing nature.
Mags is about 80 years old, making her the oldest tribute in the Quell. Actually, she wasn't originally chosen to come back into the arena. She volunteered to replace the much younger, severely traumatized woman who had been selected. (That girl is Annie, Finnick's love interest, who we don't really meet in this book.) Johanna later reveals that Mags was Finnick's mentor when he won the Hunger Games, and the two of them shared a close bond – much like the one that Haymitch and Katniss share.
Mags is one of the people Katniss identifies as a potential ally in the arena. Mags is smart and quiet. She can make great fishhooks and, once they get to the arena, is clearly an asset in figuring out what possible supplies would be safe to eat.
The problem is she's old and frail, and inside the arena that's a huge liability. When the allies have to outrun some poison gas, Mags can't keep up. It gets to the point where Katniss and Finnick can't support both her and Peeta. Rather than make the younger people sacrifice themselves for her, Mags throws herself into the gas and dies before they can stop her, absolving them of the need to take care of her.
Katniss doesn't get why that happened until later, when she wonders "[w]hy [Finnick] essentially abandoned her [Mags] to carry Peeta. Why she not only didn't question it, but ran straight to her death without a moment's hesitation. Was it because she was so old that her days were numbered, anyway?" (21.32). By the end of the book, we know that Mags was working to save Katniss on behalf of the revolution, just like the other tributes in the know were. Her sacrifice only reinforces how important Katniss' life is to the cause.