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
"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
"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)