Would visible light still be in a separate classification if we saw "colors" in a different wavelength? Basically im asking if there's anything special about visible light other than the fact that we use it to see colors. If we saw in another wavelength, would it still be possible to see colors like we do now? Does visible light have something special about it that lets us see a variety of different colors?
 A: To combine reasons given in other answers (and a comment):
Chemical reactions For us to detect light, it has to trigger chemical reactions in our photoreceptors. This less energetic the photon, the harder it is for it to react with chemicals. But if it's too energetic, it can damage our bodies. (However, there are some animals, such as electric eels, that can detect EM waves at frequencies much lower than visible light).
Interaction with water Visible light is absorbed by water less than other frequencies. This is relevant not only for aquatic animals, but also for land animals, as our eyes are filled with fluid.
Solar Spectrum You can see here that solar power peaks in the visible range. Our eyes are optimized for detecting sunlight bouncing off of objects. Other animals have sense organs for detecting other sources; for instance, the pit viper can sense infrared radiation, which is generated by living beings.
A: There is nothing special about visibile light, meaning that the electromagnetic radiation at the frequencies at which our eyes are sensibile is not intrinsically different than that at higher/lower frequency. It is just that, for some biological reasons, our eyes developed a sensibility for the electromagnetic radiation in the frequency range that we commonly define light.
It is our brain that recognizes different wavelengths as different colors, but in principle, if our sensors would be sensible in another frequency range, we would probably associate colors to other frequencies. 
A: A very special property of the visible light is that it is in a special relation to chemistry. It has the energy specter of most chemical processes.
Lower-energy light (IR) is almost incapable of inducing a chemical process of any kind.
Higher-energy light (UV) is not selective of the chemical bonds it cracks. That also gives us the upper limit of transparency for most substances (including, but not limited to, air and water).
And in the middle, there is the visible light - capable of selectively making a chemical alteration in the substance. That's why it is also possible to engineer a molecule that is sensitive to a particular wavelength (and a lot of molecules are so without any engineering effort).
So it is not only our (and most animals') vision. It is also why plants use the same visible light for their energy needs.
It is also a lucky coincidence that our star has a maximum emission in these wavelengths.
A: The range of visible light wavelengths has a special property that makes it the commonly used range for all life forms on the Earth:
It is the range of electromagnetic wavelengths that are short enough to be conveniently handled by cell sized detectors and that can pass through the atmosphere.
The Earth's atmosphere is not transparent at all wavelengths, and living tissue is also not transparent at all wavelengths.
There are small ranges ("windows") of electromagnetic wavelengths for which the atmosphere is transparent.  There is also (as far as I can tell) just one window where biological tissue can be transparent.
This diagram from the Wikipedia article on the "optical window" shows the available ranges:

Really, there are only three ranges that could be useful:


*

*300nm to 1100 nm (low UV to infrared - the visible light range.)

*Around 10 micrometers (terahertz range.)

*From around 3 cm to 12m.


Those are the ranges of "light" we can receive from the sun at the Earth's surface.
Visible light is the intersection of the wavelengths we can get from the sun and the wavelengths that can be conveniently used by biological processes.
The terahertz range doesn't pass through any living material, and the centimeter (and longer) waves are too long for convenient detection by biological processes.  That leaves only the range we call visible light.
Wikipedia has an article on the "optical window for biological tissue."  You can only make eyes out of things that will pass the light (lenses or just the pupil) and then you have to have something that will catch it and react to the absorption (retina.)
A: 
is there anything special about visible light
  other than the fact that we use it to see colors?

We can see light with wavelengths from $390$ to $650$ nm
because in our eyes we have photoreceptor cells which are
sensitive only for these wavelengths.
If the photoreceptor cells were sensitive to other
wavelengths, then we would be able to see those.

Does visible light have something special about it
  that lets us see a variety of different colors?

The special thing enabling us to see different
colors differently is, that in our eyes are
more than one kind of photoreceptor cells.
Actually we have 4 different kinds, each having
their absorbance in different wavelength ranges.
According to Wikipedia - Photoreceptor cell - Humans
and the image below there are 3 different kinds of
cone-shaped and 1 kind of rod-shaped photoreceptor cells.

(image from Wikipedia - Photoreceptor cell - Humans)
If we had only one kind of photoreceptor
cells, then we would not be able to distinguish
between colors. For example, yellow light of a
certain brightness would appear to us like red light
of the same brightness.
We would probably perceive everything
as white, black, and various shades of gray.
A: Yes, There is something very special about visible light. Each photon of electromagnetic radiation is a packet of energy that gets delivered to a single atom when the photon is absorbed. If that atom is part of a molecule, the energy that it absorbs can trigger a chemical reaction.
Your body is a big bag of chemicals, and your life processes are all chemical reactions. Photons that have sufficiently high energy can injure your tissues. You need protection from them. Your epidermis, and your hair (if you have any) help with that: They protect you from a lot of the shorter-wavelength (ultraviolet) radiation from the sun.
But your nerves and the retinas of your eye are chemical systems too. Your retinas detect light when the photons trigger chemical reactions in proteins in them.
Visible light is the range of photon energies that are sufficiently high to trigger the most delicate of chemical reactions, but not so high that the tissues of your eye would be damaged by them. So it should come as no surprise that that is the range of energies for which we evolved the ability to see.
A: Something special about the visible range is that water has low absorption in this range. 
It’s a rather sharp dip near the visible region. Since we know that life began in water, the beings that were receptive to these wavelengths had a significant advantage over the others. Thus natural selection would have favoured these life forms over the others. This maybe the reason why we are primarily receptive to the “visible” range. 
