You are, of course, familiar with the five classic senses: vision, hearing, smell, taste and touch. But when you take an elevator you can readily feel that this list is incomplete. The queezy sensation that you get in an elevator is largely due to our vestibular system, which provides us with a sense for balance and acceleration. We also have relatively distinct senses for pain, temperature, various aspects of our internal organs (interoception) and the position of our body (proprioception). But even more interesting, I think, are those senses that are completely alien to us, specialized senses that are found only in a few species of animal.
The philosopher Thomas Nagel famously wondered what it's like to be a bat (Nagel, 1974). Although being a bat must be very different from being human in a lot of ways, Nagel referred specifically to the bat's sense of echolocation, which, he felt, is so different from anything that we can experience that we can only guess how a bat perceives the world. And he has a point. Unlike nocturnal birds, which simply have very sensitive eyesight, bats listen to the echoes of their screams (too high for us to hear) to navigate through the dark. The idea behind echolocation is pretty simple: The delay between the scream and the echo tells the bat how far away an obstacle is. If the echo is heard first with the left ear, the object is on the left. And more subtle clues, such as how muffled an echo sounds, tell the bat something about the identity of the object. I would guess that, to a bat, the muffled echo of a mosquito sounds delicious.
Now you might say that echolocation is not, strictly speaking, an extra sense, because bats are essentially very good listeners. And I suppose that's true. In fact, there are people who (kind of) know what it's like to be a bat! I heard about the remarkable phenomenon of human echolocation for the first time on a Dutch talkshow called “De Wereld Draait Door”, in an interview with Tom de Witte. Tom de Witte, who became blind later in life, taught himself to “see” the world through echolocation. Unlike bats, he doesn't scream (which would be creepy), but makes clicking noises with his tongue. As you can see in the video below, the level of detail that he's able to discriminate is truly amazing (most of the clip is in Dutch, but if you skip to 2:53 you can see echolocation in action).
Video: Human echolocation
Another extraordinary sense is infrared vision. The spectrum of light that, to us, is visible, ranges from a wavelength of about 400nm (violet) to 800nm (red). Other animals are sensitive to a slightly different range, but the variation is relatively small (Land & Nilsson, 2002). The reason for this lack of diversity is simple: The sun shines brightest in the range of wavelengths that we, and other animals, have evolved to see. Being sensitive to most other wavelengths makes little sense . However, there is one other range of wavelengths that is potentially very interesting, particularly for predators, and that's the infrared range. Warm-blooded animals (i.e., mammals and birds) are usually warm relative to their environment, so they are essentially infrared beacons. Two families of snake, rattlesnakes and pit vipers, have evolved to exploit this “warm-blooded weakness” by developing infrared vision (Newman & Hartline, 1982). Unlike you might expect, snakes do not have specially adapted eyes, perhaps because the wavelength of infrared (5000nm – 30,000nm) is simply too different from that of visible light for the eyes to be of any use. Instead, snakes have dedicated “heat pits”, located under the eyes, which are anatomically very different (they are essentially open nerve endings), but functionally similar to eyes. It's still quite mysterious how snakes are able to use this infrared sense so effectively, because there is every reason to believe that the spatial resolution of this sense is very poor. Perhaps they combine infrared information with regular vision, by using their heat pits to spot potential prey and their eyes to make out the finer details.
On a final, curious note: Infrared vision is very rare, but there is at least one other animal, a particular type of beetle, which uses an infrared sense to detect fires. Amazingly, rather than flee, these "fire beetles" approach fires to lay their eggs in the burnt wood. This seems like a pretty hazardous breeding strategy, but apparently it pays off, since the beetles are still around...
 Being sensitive to a wider range of wavelengths offers little advantage, but some animals have a far more narrow range. Notably, animals that live at some depth in the ocean are sensitive only to blue light, because other wavelengths are filtered out by the water.
Land, M. F., & Nilsson, D. E. (2002). Animal eyes. Oxford University Press, USA.
Nagel, T. (1974). What is it like to be a bat? The Philosophical Review, 83(4), 435-450.
Newman, E. A., & Hartline, P. H. (1982). The infrared vision of snakes. Scientific American, 246(3), 116-127.