I always wondered how much information we get from color. Things we see have different colors; edible products change color when began to spoil so we have a notion what color a fresh product should have. And the colors we see are very small part of electromagnetic spectrum.
So is there something special in the visible spectrum or any part of electromagnetic spectrum of the same size is just as good?
Both of the existing answers provide some really nice points; I wanted to try to piece things together a little more.
The solar spectrum peaks right around the visible part of the spectrum, so it provides the most possible light to utilize - shown in this figure: 
Shows the solar spectrum both above (upper orange line) and below (shaded regions) earth's atmosphere. The visible region clearly dominated the flux - and thus this is where one would naively expect photosensitivity to develop. Note that the peak flux appearing in the visible is also tied to where earth's atmosphere has high transmittance (see also: http://en.wikipedia.org/wiki/File:Atmospheric_electromagnetic_opacity.svg).
At the same time, the infrared (IR) portion of the spectrum still offers plenty of light --- and that's part of why many animals are much more sensitive (than humans) in the IR. At night, when the sun isn't out to illuminate things, the warmth of animals themselves is enough to emit enough IR for some animals to see. It then seems reasonable to infer that our lack of IR sensitivity has to do with our diurnal lifestyle.
Finally, the visible portion of the spectrum is especially sensitive to molecular features which gives us a lot of information about materials and their differing structures and compositions. Note that this is also what allows the mechanism by which our eyes work: when a certain molecular (see: retinal) interacts with a photon, it causes a conformal change which we detect.
The visible part of the spectrum is associated with (relatively) low energy electronic transitions in molecules. The photoreceptors in your eyes operate on the basis of cis/trans isomerisation of a carotenoid molecule confusingly called retinal, where a double bond is temporarily turned into a single bond by a photon, causing part of the molecule to flip. If my memory serves me correctly, the excitation that gives rise to this isomerisation is a $\pi\rightarrow\pi^*$ transition associated with the delocalised electron cloud that results from the conjugated electronic structure of the molecule (that is, those alternating double and single bonds). Conjugated structures typically absorb strongly in the near UV and visible parts of the spectrum and are brightly coloured due to their absorption or fluorescence spectra. Metal complexes are also typically brightly coloured due to excitations involving electrons moving between the metal atom and the ligands connected to it. The vibrant red of blood and (somebody please correct me if I'm wrong) the green of chlorophyll are attributable to charge transfer bands - in the case of chlorophyll the strong absorption this molecule (actually, a family of molecules) has in the red region is pivotal to photosynthetic light harvesting. Many other pigments (carotenoids, melanin) in both plant and animal systems are involved in photoprotection - sacrifically soaking up high energy photons.
But I digress - if we go further up in energy (shorter wavelengths, into the UV region), the photons become energetic enough to liberate electrons from molecules and materials entirely. The photoelectric effect is a manifestation of this in metals. As such, ultraviolet light is considered ionising radiation and has a habit of wrecking molecules. When you go beyond this into the X-ray spectrum, you can actually knock out the most tightly bound core electrons of an atom, with disastrous effects. Gamma rays are even more energetic, and can interact with the nuclei of atoms.
If we go lower in energy (longer wavelengths) we end up in the infrared and microwave regions of the spectrum. These photons lose the ability to directly excite localised electrons (though microwaves can interact in a special way with the delocalised electrons of metals, which is why you shouldn't put metals in the microwave. Instead, these lower frequencies tend to contribute to the vibrational and rotational motion of molecules.
Now, many animals are sensitive to frequencies outside of the human visible range, in both the UV and IR regions, however the visible range, associated with fairly gentle electronic transitions, is something of a sweet spot.
There is nothing special concerning the visible light from fundamental point of view. You can have the same amount of information in any other part of the spectrum; note, however, that it has to be similar spectral interval relative to the carrier frequency. Take, for example, how much information we can transmit using radio waves.
There is, nevertheless, something special about the visible light connected to the life here on Earth. Radiation in the visible part of the spectrum is emitted most from the Sun. Therefore, it is most suitable for living organisms for gathering information about their surroundings, and that is why we see visible light with our eyes and not UV or microwaves.
