The mockingjay is a work of genius: a symbol not only of Katniss, but of her defiance against the government. The mockingjay represents a paradox: attempts to control something can have consequences you never quite intended. According to the novel, mockingjays were created when the genetically engineered jabber jays, designed to repeat what the rebels say in hidden places, mated with normal mockingbirds.
That doesn't sit well with the Capitol, which doesn't like admitting mistakes and yet can't seem to get rid of the darn things. That's a great way of describing Katniss herself: a thorn in their side who just won't go away.
One of the subtler points about The Hunger Games is that big things have very small beginnings: Katniss volunteering, for example, is going to literally change the course of history before it's all over.
The mockingjay symbol is similar. Katniss first gets a pin with the image of the bird as a gift from an anonymous old woman. It doesn't seem to mean much at that point, but it gets a lot of screen time when Katniss presents it to her sister as protection:
KATNISS: It's a mockingjay pin. To protect you. And as long as you have it, nothing bad will happen to you. Okay? I promise.
Her sister gives it right back to her before she leaves for the Games, and Cinna actually finds a way to sneak it to her just before she enters the arena. It never really leaves her after that. Little steps. Little gestures. But they grow and grow until, like the mockingjay itself, they've spread everywhere. (Even to the arena, where they literally help Katniss and Rue coordinate their attack on the Careers.)
Watch the birdie, President Snow. It's about to fly up and poke your eyes out.