Background character in The Hunger Games: Catching Fire

You are looking at the unfortunate Lavinia. She is a non-character in the movies, but does play some role in the novels.

Note - If you go to any of these links, there are spoilers for the third book. If you haven't read it, be wary. From the Wikipedia she is:

An Avox (a servant whose tongue has been cut out as punishment for treason). She has red hair, pale skin, and striking features. Because of her inability to speak, Katniss initially does not know her name or anything about her. Katniss recognizes her from when she and Gale witnessed the girl's capture and the death of her friend, and Katniss later speculates that she is a runaway from the Capitol.

To explain what an Avox is, from the Hunger Games Wiki:

An Avox is a person who has been punished for being a rebel against the Capitol; a traitor or a runaway. Avoxes have had their tongues cut out, rendering them mute. They are used as domestic servants and waiter upon tributes and Capitol citizens. It is also suggested that Avoxes are charged with the day-to-day maintenance of the Capitol and work shifts in the Transfer network under the city. The term "avox" is from Latin, meaning "without a voice."

Some more information about Lavinia, again from the Hunger Games Wiki:

Before she became an Avox, Lavinia and an unknown boy, whom Katniss believed was her brother or lover, were seen by Katniss and Gale while they were out hunting, in an off-limits wood outside of District 12. They appeared to be escaping from the Capitol for unknown reasons, and Katniss noted that the two had a "Capitol look" to them. A Capitol hovercraft appeared; the boy was struck with a spear that was shot from the hovercraft and was killed instantly, then was lifted up to the hovercraft, not to be mentioned again in the series. Lavinia, however, was scooped up by a net and taken into the hovercraft, and she briefly locked eyes with Katniss and Gale. When the boy was killed, she screamed what sounded like a name, and Katniss wondered if it was the name of the boy.

Years later, Lavinia waited on Katniss as an Avox whose tongue had been cut out for committing crimes against the Capitol. Katniss said she knew Lavinia, but at that time did not know her name or where she remembered her from. This worried the residents from the Capitol, so Peeta stepped in and claimed that Lavinia looked like Delly Cartwright - a girl from District 12 who went to school with them, and who Katniss claims, could be the nicest person on earth. As an Avox, she could not publicly acknowledge anything that was not a direct order, so Katniss' initial attempt at conversation was ignored. However when she later walks in on the mess Katniss has made in her room prior to interview night she shows some response to Katniss' attempted apology, indicating by gestures that Katniss did not need to be sorry, and that there was nothing she could have done, otherwise she too, would have been an Avox.

Finally, some trivia from the same page:

She is the only known Capitol citizen who attempted to escape for unknown reasons.

She is not mentioned in The Hunger Games film due to the thought that her presence would have been rendered useless and would not have mattered anyway.

Lavinia is also the name of a character in Shakespeare's play Titus Andronicus. In the play, Lavinia has her tongue removed after she is assaulted so she will not reveal who attacked her.

She's a truly tragic character and certainly a memorable one from the books.

_{Andrew Martin's answer gives an excellent description of this character, but without citing any canonical sources. I've found the Hunger Games Wikia to be particularly unreliable even among Wikias, so I'm going to add another answer with actual book quotes.}

She appears in the first book but not until the second film, and is used as a plot device to help us, the readers, to understand the brutality of the Capitol.

In a scene very much like the one you've screenshot (only before the 74th Hunger Games rather than the 75th), Katniss sees the Avox servant girl and recognises her:

I try to focus on the talk, which has turned to our interview costumes, when a girl sets a gorgeous- looking cake on the table and deftly lights it. It blazes up and then the flames flicker around the edges awhile until it finally goes out. I have a moment of doubt. "What makes it burn? Is it alcohol?" I say, looking up at the girl. "That's the last thing I wa — oh! I know you!"

I can't place a name or time to the girl's face. But I'm certain of it. The dark red hair, the striking features, the porcelain white skin. But even as I utter the words, I feel my insides contracting with anxiety and guilt at the sight of her, and while I can't pull it up, I know some bad memory is associated with her. The expression of terror that crosses her face only adds to my confusion and unease. She shakes her head in denial quickly and hurries away from the table.

Effie is dismissive, and through the following dialogue we (and Katniss) find out what an Avox is:

"What's an Avox?" I ask stupidly.

"Someone who committed a crime. They cut her tongue so she can't speak," says Haymitch. "She's probably a traitor of some sort. Not likely you'd know her."

"And even if you did, you're not to speak to one of them unless it's to give an order," says Effie. "Of course, you don't really know her."

But I do know her. And now that Haymitch has mentioned the word traitor I remember from where.

Peeta covers for Katniss by pretending the Avox girl bears a resemblance to someone they know in District 12. Later on, he asks for the full story and Katniss tells him how she saw the girl captured by the Capitol:

"We were hunting in the woods one day. Hidden, waiting for game," I whisper.

"You and your father?" he whispers back.

"No, my friend Gale. Suddenly all the birds stopped singing at once. Except one. As if it were giving a warning call. And then we saw her. I'm sure it was the same girl. A boy was with her. Their clothes were tattered. They had dark circles under their eyes from no sleep. They were running as if their lives depended on it," I say.

For a moment I'm silent, as I remember how the sight of this strange pair, clearly not from District 12, fleeing through the woods immobilized us. Later, we wondered if we could have helped them escape. Perhaps we might have. Concealed them. If we'd moved quickly. Gale and I were taken by surprise, yes, but we're both hunters. We know how animals look at bay. We knew the pair was in trouble as soon as we saw them. But we only watched.

"The hovercraft appeared out of nowhere," I continue to Peeta. "I mean, one moment the sky was empty and the next it was there. It didn't make a sound, but they saw it. A net dropped down on the girl and carried her up, fast, so fast like the elevator. They shot some sort of spear through the boy. It was attached to a cable and they hauled him up as well. But I'm certain he was dead. We heard the girl scream once. The boy's name, I think. Then it was gone, the hovercraft. Vanished into thin air. And the birds began to sing again, as if nothing had happened."

"Did they see you?" Peeta asked.

"I don't know. We were under a shelf of rock," I reply.

But I do know. There was a moment, after the birdcall, but before the hovercraft, where the girl had seen us. She'd locked eyes with me and called out for help. But neither Gale or I had responded.

So that's the tragic story of Katniss and the Avox girl. The name Lavinia, I believe only appears in the second book.

Interestingly, she comes originally from the Capitol, not one of the Districts like (I assume) most 'traitors'. And it seems likely, in view of the revelations in the second and third books, that they were heading for the secret District 13 - and nearly got there, as well:

"They were from here?" he asks, and he secures a button at my neck.

I nod. They'd had that Capitol look about them. The boy and the girl.

"Where do you suppose they were going?" he asks.

"I don't know that," I say. District 12 is pretty much the end of the line. Beyond us, there's only wilderness. If you don't count the ruins of District 13 that still smolder from the toxic bombs.

_{(all quotes from The Hunger Games, Chapter 6)}