Is it possible that there is a color our human eye can't see? Is it possible that there's a color that our eye couldn't see? Like all of us are color blind to it.
If there is, is it possible to detect/identify it?
 A: The eye is sensitive to light with a wavelength in the range from about 700nm to 400nm, and for the non-colour blind all wavelengths in this range are detected by one or more of the cone cell types. So there are no hidden colours in this range.
Light outside the 700-400nm range can't be seen, so I suppose you could claim these are hidden colours, but then we tend to define the word "light" to mean what we can see, and we'd say the wavelengths larger than 700nm are infrared and those smaller than 400nm are ultraviolet.
Actually, it's alleged that if you have the lens of the eye removed (it can happen due to eye problems) then you can see further into the UV. This is because the lens absorbs UV light and when it's removed that light can reach the retina and be perceived. Maybe this counts as a hidden colour. I feel disinclined to try the experiment :-)
A: Color is basically formed in the brain not the eyes. Also the human eye can handle electromagnetic waves from 4000 to 7000 Angstrom, roughly, so called visible light. Above this range, the infrared region is found. It is not red in color or something, it is a name convention. Our eye can not handle it and so the brain dose not recognize it.
It is complicated if you are thinking it for the first time and can be extremely messy. 
So color dose not exist its different from species to species.
A: There are different types of color blindness. 
In color vision tests (patches of color when you can see digits or not) there are some tests where people with normal vision can't see the figure, but people with a specific color blindness can see it. That means people with normal vision are color blind to some specific color differences.
It doesn't mean this color will appear gray to you. It means two patches will seems the same color to you (if you have normal color vision) and can be distinguished one from another by someone else (who is supposed to have a bad color vision)
If you use a spectrograph, even in only the visible wavelength range, you have much more data (the proportion of each wavelength) than you can have with a normal human eye, that summarize it to only three values.
A: There's been found some women who are quadrachromatic however they're very rare but in comparison to them we all are colourblind as they can see hue's that we can't see.
A: I'll add this to the awesome answer from Chris White:
People with synesthesia may experience color when stimulated by other sensations, like sounds or letters for example. And some such people have reported seeing "alien colors" that only exist in their visual field when they look at certain graphemes, like punctuation.
It is certainly possible that such "alien colors" may indeed be perceived, and yet be impossible to reproduce in the physical world (by combining visible frequencies) precisely because they are the result of direct/internal neural stimulation, and are not constrained by the same rules which hold true for the neural signals generated by the color receptors in human eyes.
If this is true, then it is also possible that we will someday be able to detect, record, and reproduce such "alien colors" when we learn enough about how human visual processing works to be able to build high quality artificial eyes.
A: It really depends on what you mean by colour.
If by colour you mean "the human brain's response to a given combination of wavelengths", then by definition there can be no invisible colours; wavelengths combinations that do not stimulate any cones in the eye are just equivalent to black.
If by colour you mean "a given combination of wavelengths", then we are actually totally blind to almost all of them because light is a multidimensional signal, and our eyes can only grasp three to four dimensions out of these. For instance, we are unable to tell the difference between a pure 550nm (what we see as "green") wave and a combination of 520nm and 580nm waves; certainly they are different signals, yet our visual system makes us believe they're equivalent.
If by colour you mean "a single, unique wavelength", then we can actually see colours that do not exist; for instance, there is a single wavelength for orange (around 620nm), but there is none for purple (which is invention of our brain to describe combinations or red and blue).
A: Quickly, try this: Imagine blindingly bright red light! Now blue! Now yellow!
You could see stark differences as you shifted from color to color, couldn't you?
Yet if you think about what just went on inside of your head, it didn't involve any color photons going into your eyes, did it? So, what you just did must be separate from the light frequencies picked up by your eyes. The fact that you could easily distinguish between each of those in-your-head-only phenomena shows they are physically meaningful phenomena. The fact that they are complicated, low-energy, poorly understood phenomena that only operate within your brain doesn't make them any less real, just a lot harder to access and analyze.
The more philosophical term for these in-your-head-only phenomena is qualia (Kwal ee ah). We tend to assume that all humans share the same qualia for light, because we have uniform labels for the bands of light that evoke them.
However, the strong form of that assumption is almost certainly incorrect. There is for example a wonderfully odd condition some people have called synesthesia, in which sensory inputs get mixed up and mapped into multiple qualia. Mostly it involves color being added to letters and numbers, but in some of the more radical forms touching a certain spot on someone's leg can evoke a color or a smell.
Even for those of us who don't have synesthesia (I'm extremely jealous of those who do), qualia can get remapped. I once lost my sense of smell for a while, and when it came back, the first two smells I encountered (only) would up remapped into entirely new qualia. Consequently, second-hand cigarette smoke and gasoline now both smell like edible foods to me (yuck!). That was emphatically not the case before my brain decided to "remap" the signals they evoke chemically in my nose.
So, putting all of that together, the answer to your question is twofold:


*

*Are there light spectra that some creatures can see but humans can't see? Definitely yes, since for example there are birds that have receptors for four light bands instead of just three. Their extra receptor is in what we would call ultraviolet. (Their other color receptors are also not quite the same as ours.)

*Are there qualia that some creatures can see, "in their heads only," that humans cannot imagine? This question is trickier than it looks, because at present there is no technology that can be used to detect the apparently subtle differences between qualia in a functioning brain. My best guess is that it's quite likely that birds that can see in the ultraviolet also have a unique quale ("Kwal ay", the singular of qualia) that helps them interpret their larger range of sensory inputs. So, probably they see something different.
We don't know that for sure, however. For example, it could be that such birds simply stretch the same qualia we use when imagining a rainbow to cover a broader range of light spectra. In that case, ultraviolet to a bird would just look the same as what we call violet.
So why do I think such birds have a unique quale to represent ultraviolet light?
Well, mostly because of this: Assuming you are not color blind (my apologies about this one if you are): Imagine red! Imagine green! Did those two qualia look very similar to you? So much so that you have trouble recalling which is which? No? Not at all? In fact, some of you are likely right now screaming in your heads, "You nincompoop, red and green qualia don't look anything alike! How could you ever even think that?"
Well, very easily if I was red-green color blind. You see, what most people don't realize is that red-green colorblindness is the norm for all mammals except primates.
Primates picked up an extra light-sensing protein mainly because they eat a lot of fruit. Fruits, however, have a curious property called "ripeness" that on average they tend to advertise by undergoing some kind of color change. The most common such change is to go from green (not ripe) to red (ripe). Unfortunately, mammals in general can't see this particular color change, which places a dog for example at a distinct disadvantage if it is hungry and trying to find ripe fruit as a fallback food source.
So to handle fruits better, primates have this extra sensory protein for green light, one that is structurally derived from and still remarkably similar to the red-sensing protein that all mammals have.
But here's the critical point: We did not just get another color sensor, we also got a new, starkly different quale (imagine green!) to go with it. People without red-green color blindness would tend to agree that this new "it's not a ripe fruit" quale is quite distinct from the older red quale (imagine red!) that previously included that same turf.
That strong distinction between two qualia helps us transform spectra differences that our eyes see into a real survival advantage, specifically by making it trivial and fast to look over a tree and notice red fruits standing out like sore thumbs. Some mushy slight difference, like that between some shades of blue, would not be nearly as effective for this quick sorting-out process.
So: If a bird adds in ultraviolet protein receptors, wouldn't it make sense that they would also have a new quale specifically to make that extra sensory input stand out? That's why my bet is that birds whose eyes have receptors for ultraviolet light also see ultraviolet as a new color quale, that is, as a completely new color sensation that we humans quite literally cannot imagine.
So, to wrap it up: What are qualia?
No one has the foggiest idea! Sorry.
But my hope is that someday through methods like fMRI, we will actually begin to understand what is going on in the brain well enough to detect when different qualia are in action. Then and only then we may gain the ability to know for sure whether my in-my-head-only definition of "red" really does match the one inside of your head.
And even further down the pike, who knows? Simple electrodes can certainly evoke powerful sensations -- qualia -- within the human brain. Perhaps someday someone will figure out some clever ways to convey the birds-only quale for "ultraviolet" into the brain of a human volunteer. That lucky person would then get to see, for the first time in human history, a color that no one has ever seen before, one to which the entire human race has been quite literally colorblind for its entire prior existence.
Now wouldn't that be a wonderful thing to behold?
A: As mentioned in a number of other answers, there are three different color receptors in a typical person's eye. They respond to different wavelengths of light, as can be seen in the below diagram from wikimedia.

The $x$-axis is wavelength in nanometers, and the three curves represent the three receptors' response at those wavelengths. Any incoming light will affect each of these to a certain degree. Thus the range of theoretically perceivable colors is basically the set of all different triplets of response values for these receptors. (Think "blue is at 25%, red is at 97.3%, green is at 12%.") When all three are firing near full strength, the result is something like white. If the blue receptor is firing and red and green are basically off, well then you see blue.
There are two important points to make, though. First, one often sees reference to a connection between wavelength and color. Indeed, you cannot see any wavelengths outside approximately 400 to 700 nanometers. [Note that other animals have different ranges: Bees can see into the ultraviolet (below 400 nanometers), while some snakes can "see" into the infrared (above 700 nanometers).]
Be careful not to take this connection too far, however. In particular, there is more to color than a single wavelength. For instance, light could be hitting your eye with two overlaid wavelengths - one of which resonates with the green receptor very well, and the other of which resonates particularly well with the blue. The resulting perception is likely to be a teal that simply cannot be reproduced with a single wavelength. This is exactly analogous to sound, where a monochromatic "pure" pitch will never, at any frequency, sound like a trumpet or a viola - those instruments' timbres are defined by the varying strengths of the overtones. In other words, "all the colors of the rainbow" does not encompass all colors.
The other point is that there are valid combinations of receptor stimulation levels that cannot be achieved by any combination of wavelengths. This is partly due to how your receptors' ranges are not separate. Note for instance how the "red" (L) and "green" (M) receptors are actually quite close. It is hard to stimulate one without the other. You can never, for example, get "100% green, 0% red and blue" as a signal from your eye to your brain. Such theoretical colors that cannot be reproduced with any source of light are called imaginary colors. Supposedly, you can actually see some imaginary colors by first saturating one or more receptors (say by looking at nothing but lots of pure green for a few minutes), thus wearing them out, and then looking at another source of light. The response you get won't be quite the same as you normally would with that light source, since some of your receptors are not up to full capacity. (I have not had too much luck with this experiment myself, but perhaps you may fare better.)
Finally, regarding detection: When it comes to light, all it is scientifically is different wavelengths of electromagnetic radiation. We have spectrometers for pretty much every wavelength out there, well beyond visible. Thus you can always tell the exact composition of some light ("12% in the 550-553 nanometer range, 80% evenly distributed between 600 and 700 nanometers, 8% focused at 350 nanometers," for instance). We don't need to rely on our eyes' physiology.
A: We have color perception because we are trichromats. In our genes there is code for three slightly different light-sensitive molecules. The light-sensitive cells in the retina are called cones, and neighbouring cones each produce one of the different versions of the light-senstive molecule. So each of the three cone-types responds slightly differently to the incoming light, and then neuron cells compare these responses. 
The pixels of our computer monitors and our television sets come in three colors. Just three colors. Those three colors are sufficient for satisfactory color reproduction. The reason three colors are sufficient is that our eyes have just three types of cones.
In evolutionary history trichromacy is a relatively recent development. Primates are trichromats; many mammals are dichromats. If we humans would all be dichromats then our computer monitors and television sets would need just two colors for satisfactory reproduction of all the colors we can see. (EDIT - Sam Hocevar has pointed out in a comment that the statement about color reproduction is oversimplified.)  
So it's a matter of how many different light-sensitive molecules are available, and how well the neurons do in comparing the responses from differently sensitive cones.
We trichromats have access to a bigger color world than dichromats have. There are colors that to a trichromat look different that are identical to a dichromat.
Conversely, a species that is tetrachromatic (and with neuron wiring to compare all different responses) would have access to a yet bigger color world.
Compared to a fully functioning tetrachromat we trichromats are partially color blind. 
A: I haven't provoked this in original answer because it would have been extremely messy, but now i have to.
 Your brain receives signal for 520 nm(5200 Angstrom), now you have told bye your teacher or parents that this particular type of signal is green colour hence you see a leave of tree as green, what if since birth you have been in other world and you have been told 520 nm as Red and instead of green and vice versa, you would have always thought as yo have been said. then for you Leaves would have been Red and Strawberry Green. It doesn't matter. is it? All that matter we can identify 520 nm and 660 nm as different wavelength, its up to us what to name it.
A: This is an old question, but I'm very surprised no one has mentioned this:
You CAN'T see red-green.
According to multiple websites, some studies showed that the human eye cannot see both red and green simultaneously, as the red cone and the green cone send out signals that cancel each other out.
Similarly, blue-yellow is equally impossible to perceive.
There you have it. Two "colors" that is a mix of the colors of the rainbow, yet are impossible to detect. They are known as the FORBIDDEN COLORS.
A: Of course it's possible, e.g. the colors seen by bees are invisible for us.
