In a nutshell, the problem of consciousness is the problem of trying to explain what consciousness is and how it is produced in the brain.
When we look at a piece of red fabric, for example, all sorts of electrochemical reactions ensue causing us to perceive red, but that perception, the experience of redness, is not an electrochemical reaction, it's not physical at all. No one examining the brain of a person looking at the fabric would see red anywhere in the brain, so what exactly is going on when we see a color, or smell a fragrance, or hear a sound? No one knows.
This is called the "Hard Problem" of consciousness (as opposed to the easier problems such as mapping which parts of the brain are active when we have various experiences). The Hard Problem is explaining what those experiences are in the first place.
The Guardian piece is a little long, but it gives some interesting background on how the problem gained the attention of philosophers back in the 1990s. Here's the lede:
One spring morning in Tucson, Arizona, in 1994, an unknown philosopher named David Chalmers got up to give a talk on consciousness, by which he meant the feeling of being inside your head, looking out – or, to use the kind of language that might give a neuroscientist an aneurysm, of having a soul. Though he didn’t realise it at the time, the young Australian academic was about to ignite a war between philosophers and scientists, by drawing attention to a central mystery of human life – perhaps the central mystery of human life – and revealing how embarrassingly far they were from solving it.Not all philosophers are as impressed with this problem as Chalmers is, of course. Those philosophers who are materialists (who believe that everything is reducible to matter) don't think there's a problem here at all. They argue that your sensations, ideas, self-awareness, etc. are all just the product of neurons firing in certain ways. They may be right, but they have no explanation for how the physical, material phenomena of an electrical current flowing along a nerve and transmitted across a synapse by molecules results in the experience of pain. Indeed, it's hard to see how the sensation of pain can be simply a material phenomenon.
The scholars gathered at the University of Arizona – for what would later go down as a landmark conference on the subject – knew they were doing something edgy: in many quarters, consciousness was still taboo, too weird and new agey to take seriously, and some of the scientists in the audience were risking their reputations by attending. Yet the first two talks that day, before Chalmers’s, hadn’t proved thrilling. “Quite honestly, they were totally unintelligible and boring – I had no idea what anyone was talking about,” recalled Stuart Hameroff, the Arizona professor responsible for the event. “As the organiser, I’m looking around, and people are falling asleep, or getting restless.”
He grew worried. “But then the third talk, right before the coffee break – that was Dave.” With his long, straggly hair and fondness for all-body denim, the 27-year-old Chalmers looked like he’d got lost en route to a Metallica concert. “He comes on stage, hair down to his butt, he’s prancing around like Mick Jagger,” Hameroff said. “But then he speaks. And that’s when everyone wakes up.”
The brain, Chalmers began by pointing out, poses all sorts of problems to keep scientists busy. How do we learn, store memories, or perceive things? How do you know to jerk your hand away from scalding water, or hear your name spoken across the room at a noisy party? But these were all “easy problems”, in the scheme of things: given enough time and money, experts would figure them out.
There was only one truly hard problem of consciousness, Chalmers said. It was a puzzle so bewildering that, in the months after his talk, people started dignifying it with capital letters – the Hard Problem of Consciousness – and it’s this: why on earth should all those complicated brain processes feel like anything from the inside? Why aren’t we just brilliant robots, capable of retaining information, of responding to noises and smells and hot saucepans, but dark inside, lacking an inner life? And how does the brain manage it? How could the 1.4kg lump of moist, pinkish-beige tissue inside your skull give rise to something as mysterious as the experience of being that pinkish-beige lump, and the body to which it is attached?
Anyway, there's a lot more on this fascinating topic at the link.