The Bone
No bones about it; this is an important symbol in 2001: A Space Odyssey…. Sorry about that, we couldn't help ourselves.
Horrible puns aside, it's an important symbol. The bone symbolizes humanity's tools and our development of technology.
After contacting the monolith, an early hominid sits on top of a pile of tapir bones. He then toys around with a bone, giving the other bones a few experimental taps. He then wallops the skull hard enough to shatter it into pieces. This hominid has become the first human inventor, and he's invented the club.
This new invention converts momentum into power, allowing him to strike harder than he could with his hand. It also extends his reach and prevents the risk of injury that attacking with only his body would expose him to. It's no wheel, but it's not bad for the first technological invention on Earth.
Using this new tool, the hominids begin to act more like humans than we've previously seen them. They hunt tapirs, giving them meat which will increase their caloric intake and further allow for the development of their brains. They also begin to walk around more upright.
Finally, they take their new weapons to the watering hole and chase off rival hominids, displaying their power by beating the rival group's alpha male to death.
The implication is that it is our development of tools and technology that led humanity to evolve. This development allowed us to conquer our planet. The bone club eventually led to other inventions such as the tools of agriculture, hunting, travel and architecture. Of course, as evident in the watering hole scene, this also led to not-so-great developments like war and the Cat Translator.
Throwin' Dem Bones
Still stoked over victory, one of the early hominids, presumably Moonwater, tosses his bone club into the air. The club spins and spins and spins until we are given an unexpected match cut: as the bone club slowly drops back down, it becomes a similarly shaped satellite orbiting Earth.
This is perhaps the most famous match cut in cinematic history, and through it we jump 4 million years in the future in an instant. No complicated rules, no sling-shot maneuvers around the sun, no gunning it to 88 mph; we're just there.
The match cut is important to understanding the bone's importance in the film. The film is suggesting that all of our technological advancements came from this makeshift club. Just like we evolved from Australopithecus afarensis, and war evolved from waterhole skirmishes, so do satellites have their origins in that original technology.
And just as the club improved the lives of our hominid ancestors, technology helps the astronauts survive in space in 2001. In time, our ancestors would use tools to escape the confines of Africa and invent their way to the top of the Earth's food chain. In the same way, the humans in 2001 will develop tools to help them escape the confines of Earth and invent their way to the moon and beyond.
Director Hugh Hudson said of Kubrick: "We worship technology now. And he predicted that somehow. He saw where we were going in the 60s. I mean it completely controls our lives." (Source)