The title refers to the annual blood-sport that Katniss gets pulled into: 24 Tributes from the 12 Districts get dropped into an arena and battle to the death to prove that the Capitol is still in charge.
But there's a lot going on in this title. For starters, there's the word "Hunger," which implies suffering and starvation, and comes to mean not only what those lucky kids in the arena get to go through (knife-throwing psychopaths is only one of the amazing variety of ways they can die), but also the suffering in the various Districts. In fact, the winning Tribute's District gets extra food as a prize.
The second word in the title, "Games," demonstrates that the Capitol literally makes a sport out of that suffering, broadcasting it every year like some demented Super Bowl. It trivializes the deadly situation that the Districts are in.
Sums up the whole vibe of the movie pretty well, doesn't it?
Beyond that, the name is intended to evoke the feeling of Ancient Rome at its worst: pageantry and spectacle covering up the kind of bloodletting usually reserved for movies with the words "massacre" in the title. That's by design. Author Suzanne Collins named her country Panem, which is actually a Latin word meaning "bread." It usually starts the phrase "bread and circuses," which is the way Ancient Rome kept a lid on poverty and other social problems. Keep them entertained and they'll happily put up with whatever injustice you want to cram down their throat (source).
But that doesn't have any bearing on the world we live in today, does it?