Mary is a scientist who has specialized in Vision and Colors. She knows everything there is to know about colors and vision; how we see objects, how light gets reflected, what neurons in the brain decipher the perceptual signals, what are colors, their wavelengths, their nature, their combination, etc. Anything you need to know on colors and vision, Mary has the answers. Yet, Mary herself is an achromat - she has no color vision. She views the world in just two colors; black and white. The cones in her eyes are not defective themselves; the part of her brain responsible for processing the color signals are defective. Of course, with advances in neuroscience and surgery, this can be fixed. Mary is all set for her surgery, curious to know how the world would look, bathed in all the colors she has meticulously studied. Hmmm... why would a scientist who has expansive knowledge of all colors, wonder how the colors would look like to her? Seems like despite knowing everything there is to know about colors, she still doesn't know quite a few aspects about these colors... (Source: "What Mary didn't know", by Frank Jackson)
This is quite a classic hypothetical scenario used by dualists (those who believe in two entities- mind and body, interacting with each other) to spark a debate with the physicalists (those who believe in only the existence of a body and try to explain/reduce mental states through physical states). I am a dualist (else I wouldn't be using the word Mind all over my blog;)). I do believe in an entity that is separate from a physical body, and assert that not all mental states can be reduced to physical states. Therefore I'm sure to bring in my biases as I put forth my views, but argue with me to help me question my biases :)
It's one thing to know about the physical attributes of perceptual stimuli but it's another aspect to feel them for yourself. Mary might know everything about the color green, yet when she sees a palette of colors, she needs her sensory stimuli to know which color is green. Beyond the work of her sensory stimuli (which is reducible to physical states), is a mental state - a feeling that is associated with the color green. This feeling is impossible to be measured and is highly subjective. This subjective qualitative experience of perceptual stimuli is called qualia.
A related aspect is the inverted spectrum problem. My visual experience of looking at green, might be your visual experience of looking at the color red. This can extend to many other experiences. Pain, happiness, taste, smell etc. I can read everything about a vegetable I have never tasted - I might have heard that it tastes sweet, crumbly and hard to chew... yet I wouldn't know the exact taste of the vegetable, and neither would I know what sweet and crumbly are, unless I've experienced them. I can ask millions to gather more adjectives, yet the essence of the taste cannot be verbalized, neither can it be expressed in physical terms. It is a combination of sensory experiences, that are quite subjective. With advances in Neuroscience, one can probably understand the neurons that fire when people eat an apple and can compare that against thousands of samples to statistically prove similarities in patterns. Yet, there is noway for us to prove that the quality of feelings associated with the neurons are the same.
Consider another example - we measure the activity of neurons when a standard bob pin is poked (using a uniform pressure) into 100 individual's arms (let's make this experiment clean by considering a homogeneous sample of same age, same, gender, same weight). The pain measured via the neurons being stimulated might look similar, yet if we are to ask the participants' rating of their pain, it is going to be highly subjective! There is no way to verify person X's feeling of the pain, with person Y's feeling of the pain! Don't they say it's all in the mind?! There are some brave souls out there going through surgery without anesthesia (in the name of hypnosis, but that is a tangential topic).
There is active research on transferring tactile sensations from expert to novice through haptic devices that try to replicate and simulate force-feedback sensations as experienced by the experts. Yet only a diluted extent of parameters are feasible to be measured (and precision is another issue). But let me not get too emphatic. With progress in neuroscience and haptics , maybe qualia can one day be expressed in physical terms. But the impossibility of verifying qualia against different people (unless we transpose from one body to another), still leaves the stone unturned.