In The Hunger Games: Catching Fire, there is a three finger salute that is constantly shown, sometimes with whistle sound. But what does this three finger salute mean?
From Chapter 2 of the first book:
To the everlasting credit of the people of District 12, not one person claps. Not even the ones holding betting slips, the ones who are usually beyond caring. Possibly because they know me from the Hob, or knew my father, or have encountered Prim, who no one can help loving. So instead of acknowledging applause, I stand there unmoving while they take part in the boldest form of dissent they can manage. Silence. Which says we do not agree. We do not condone. All of this is wrong.
Then something unexpected happens. At least, I don’t expect it because I don’t think of District 12 as a place that cares about me. But a shift has occurred since I stepped up to take Prim’s place, and now it seems I have become someone precious. At first one, then another, then almost every member of the crowd touches the three middle fingers of their left hand to their lips and holds it out to me. It is an old and rarely used gesture of our district, occasionally seen at funerals. It means thanks, it means admiration, it means good-bye to someone you love. [-Katniss' Narration]
So it's effectively used to say goodbye.
In the first book, when little Rue dies, the following scene takes place:
A few steps into the woods grows a bank of wildflowers. Perhaps they are really weeds of some sort, but they have blossoms in beautiful shades of violet and yellow and white. I gather up an armful and come back to Rue’s side. Slowly, one stem at a time, I decorate her body in the flowers. Covering the ugly wound. Wreathing her face. Weaving her hair with bright colors.
They’ll have to show it. Or, even if they choose to turn the cameras elsewhere at this moment, they’ll have to bring them back when they collect the bodies and everyone will see her then and know I did it. I step back and take a last look at Rue. She could really be asleep in that meadow after all.
“Bye, Rue,” I whisper. I press the three middle fingers of my left hand against my lips and hold them out in her direction. Then I walk away without looking back.
So Katniss makes this symbol and as the movie shows, the first riots and protests of the uprising against the Capitol begin.
From then on, the significance of the gesture completely changes and it becomes a symbol of the revolution - effectively, a way for the citizens of the Districts to say goodbye to the overbearing force and power of the Capitol.
The whistle that accompanies it is the noise that the mockingjays make. From the Hunger Games wiki:
A mockingjay is a bird that was created through the mating of jabberjays and female mockingbirds. When Katniss wore a pin with this bird on it in the 74th Hunger Games, the Capitol was angry. They believed it would result in an uprising because the Capitol saw the mockingjay as a failure. This is why Katniss became "The Mockingjay" and led the rebels in Mockingjay.
The salute is shown more in the movie because it’s an action that is loud and clear. As they say, "actions speak louder than words"; and as they didn’t have Jennifer narrate the film, then they had to show more actions. Otherwise, the audience would be more in the dark than Katniss was throughout the whole series.
The three finger salute is said to be a farewell to a loved one. That is how the people in the districts see it. Those who serve the Capitol and Pres. Snow see it as an act of rebellion of sorts. Those who put the three fingers in the air and do the whistle are showing signs of anti-Games beliefs.