The District 12 sign is used by District 12 residents when they have to say thanks or just to show that the person is loved and respected by them. It's a gesture of admiration, gratitude and saying goodbye to someone you love.
The sign is made by pressing your three middle fingers of your left hand to your lips and then hold them out to the person, or people, that you want to show respect to. In the books, the left hand is used, but in the first film it is the right hand, and the fingers are simply held up by their side. Later on in the second film, the left hand is used like in the book.
The Hunger Games
Katniss Everdeen became District 12's tribute at the reaping of the 74th Hunger Games by volunteering to go into the arena to save her younger sister Prim. As Katniss was walking up to get on the stage, Prim grabbed onto her and needed to be pulled off by Gale. When Katniss was standing on the stage, the rest of District 12 were asked to applaud, but instead they used the respect sign to say goodbye to her.
When Rue had been killed by a spear in the stomach by the boy from District 1, Marvel, Katniss had shot him in the upper chest with one of her arrows. To show the Capitol that she was "not just a pawn in their games," Katniss decorated Rue's body with flowers which the cameras would have to show when the body was collected. Just before the hovercraft came, she used the respect sign for Rue.
In the film, after Rue's death Katniss showed the respect sign to the cameras, the people of District 11 did the same, which sparked the uprising of District 11.
Quarter Quell Reaping
The Victory Tour
In District 11 during the Victory Tour, once Peeta and Katniss had finished their speeches, which included the personal speech and the speech said in every district, the crowd used the District 12 respect sign for Katniss, which was the last goodbye that she gave to Rue in the arena. The old man that had started it during the speech in District 11 was then publicly killed.