The monolith appears in every section of the film, but we never get a full explanation of its meaning. It's a massive, tall black rectangular object. It looks like a superbig external hard drive. Which it kinda is. One with all the information of the universe on it—we're guessing that's, like, zillions of petabytes.
Enigmas from the beginning, the monoliths are still mysterious after the credits have rolled and the Blu-Ray extras have been thoroughly sleuthed for clues. It's not elementary, dear Shmoopers.
Floyd's pre-recorded message to the Jupiter crew and his discussions with his moon-based colleagues provide us with the only real clues we have:
FLOYD: Eighteen months ago, the first evidence of intelligent life off the earth was discovered. It was buried 40 feet beneath the lunar surface near the crater Tycho. Except for a single, very powerful radio emission aimed at Jupiter, the 4-million-year-old black monolith has remained completely inert, its origin and purpose a total mystery.
From this, we can say with certainty that the monoliths were created by extraterrestrials, and… that's really where the certainties end. The moon monolith is a signaling device, but the Jupiter monolith serves as a Star Gate and the prehistoric one as a genetic transmogrifier. We can't even say what the monoliths do—are they signal devices or multi-taskers?—or how they do it.
So how do we solve a question that the movie itself doesn't answer? We don't really. Instead of answers, we'll have to deal with possibilities we can infer from the film's narrative and imagery.
Monolith ex Machina
As Carolyn Geduld has noted, "The easiest sensible interpretation of the slab […] is to call it a religious symbol. Kubrick, however, has pointed out that alien technology would probably look strange enough to appear godlike to humans on Earth" (Source).
There's a good amount of evidence to back up backup Geduld's reading. György Litgeti's "Requiem" plays when the monolith is discovered by the early hominids as well as when Floyd reaches it on the moon. This score imbues these scenes with a sense of awe, and the name of the work refers to a type of song sung during Catholic Mass.
Both the hominids and Floyd treat their respective monoliths with a type of religious reverence. The hominids are initially terrified—as most characters tend to be when faced with the almighty—but they soon gather around its base in a huddled group that draws parallels to bowing or kneeling motions used in worship. When Floyd touches the monolith, his hand does so slowly and deliberately, as though he's admiring an object with totemic power rather than studying an object with detached objectivity. Finally, when the monolith appears before Bowman on his deathbed, it does so in a dominating position that resembles an angel of death.
Like a religious artifact, the monolith also appears supernatural. Now, supernatural doesn't necessarily mean "magic"; it more broadly means anything that is unexplainable by natural laws. As noted in Geduld's quote above, these monoliths likely follow natural laws that are so beyond our understanding of nature that they appear magical.
For example, its ability to change the hominids into tool-wielding humans appears to be a supernatural manipulation of nature. This is similar to the way people believe some religious relics have the ability to heal the sick beyond what seems possible scientifically.
We can claim that the monolith is a wonder of science and technology, but given our lack of understanding, we would be just as correct to say it works miracles.
The monolith points to entities that are so powerful they might as well be gods compared to humans. As Kubrick pointed out in an interview on the 2001 Blu-ray:
Can you imagine the evolutionary development that much older life forms have taken? Their potential would be limitless. And their intelligence ungraspable by humans. These beings would be gods to the billions of less advanced races in the universe. They would possess the twin attributes of all deities: omniscience and omnipotence. They would be incomprehensible to us except as gods.
The film plays with this idea and gives the unseen ETs qualities we'd associate with God. Perhaps the most obvious is the fact that they remain unseen throughout the film, yet like a God, they guide humanity toward the heavens where they dwell (read: outer space).
As noted above, the extraterrestrials appear omnipotent—that is, all-powerful. They can travel vast distances of space-time, construct star gates, and manipulate natural phenomena. And they don't have any physical limitations, such as death, that we know about. They don't even seem to have bodies.
Omniscience means to know everything there is to know. Their knowledge is light-years beyond ours, as evident by their seemingly omnipotent abilities. They're basically the Alex Trebeks of the universe, minus the cheat sheets.
Their role in the film also seems to be to create humanity in their image. They alter the evolutionary path of the hominids towards that of homo sapiens, and later they change Bowman into a Star Child. This final evolutionary state shares at least one trait similar to the extraterrestrials: the ability to traverse space like it owns the place. This is a nod to several religious beliefs that claim God (or the Gods) created humans to be in his image, such as in Genesis 1:27.
The result is a type of scientific mysticism where science reaches such an advanced degree that it seems to us like magic would have seemed to early humans—i.e. we can't explain it except to say that it works. If you're interested in exploring this concept further, several of Arthur C. Clarke's other works, such as Childhood's End, contain similar themes.
But Wait, There's More
In her analysis of the film, Geduld notes several other examples of ways to interpret the symbol of the monolith.
She points out that it can be viewed as a symbol of technology in general. Like HAL aboard the Discovery One, the monolith controls humanity to the point that humanity couldn't survive without it. Without the monolith, the hominids don't invent the bone club, they don't evolve into humans, and we become leopard chow. With this reading, it becomes a symbol of the benefits of technology but also represents our complete reliance upon it.
It can also be seen as a symbol of predestined fate. The monoliths lead humanity to Jupiter like some big black breadcrumbs. Also, it isn't like the early hominids or Bowman decided they wanted to become the fulcrum for the next stage in humanity's evolution. For that matter, it isn't like they decided what stage that development would take; for all we know, Bowman would have preferred to be Adonis. Instead, the extraterrestrials shaped us in the image they wanted, resulting in an atmosphere of determinism that hangs over the entire film.
Like the world's most simple Rorschach test, the monolith symbolizes different things to different people. Just don't say it's a butterfly; that's cheating.