Is it true that animals see colours?

- Animals see in colors thanks to a type of specialized cells located in the retina, the cones, that are sensible to the colors of the light. Humans have around 6-7 millions of those cones (and now imagine how did they manage to count them). tl; dr different type of cones respond to different wavelengths, we humans have 3 different types of cones, the majority about 64 percent of them respond most strongly to red light, while about  a third are set off the most by green light. Another 2 percent respond  strongest to blue light. (Source: How Do We See Color?). So, basically we see in red, green and blue, 3 primary colors, and the rest of them all is got by mixing these 3.

Source Taringa.net.
- The small and grumpy purple spot mantis shrimp, Onodactylus smithii, is the winner of vision in colors.
Remember we have 3 primary colors?. Well, this critter has 11 or 12 primary colors. Also, they have eyes that simultaneously measure four linear and two  circular polarisations, enabling them to determine both the direction of  the oscillation, as well as how polarised the light is. (Source: Page on sciencedaily.com).

Souce of this beautiful image: http://www.nature.com/scitable/b...
- Many birds see ultraviolet.

IN THE EARLY 1970s, A RESEARCHER testing the ability of pigeons to discriminate colors discovered by accident that the birds can see ultraviolet (UV) light.  The finding was deemed curious but not too important. “It was natural  for scientists to assume that bird vision is like human vision,” says Geoffrey Hill, an Auburn University ornithologist and the author of Bird Coloration.  “After all, birds and humans are both active by day, we use bright  colors as cues. ... No one really imagined birds might see the world  differently.”
But during the following decades, systematic testing of bird vision  revealed something unexpected: Many bird species—not just pigeons—can  see UV light. Indeed, with the exception of night-flying birds such as  owls, the eyes of most birds probably are even more sensitive to  ultraviolet light than they are to what we call visible light.  Scientists also have learned that many birds have plumage that reflects UV light. Together, these discoveries “made us realize there could be new answers to old questions,” says Drake University biologist Muir Eaton.  Birds rely on vision to choose mates, find food and scan for predators,  for example. “If you assume birds see exactly what we see, you could  have the wrong framework for understanding bird behavior,” Eaton says.

Souce (True Colors: How Birds See the World - National Wildlife Federation)

Also, many insects are able to see ultraviolet. This time you'll have to trust on me because I'm not adding links here.