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A broadly acceptable definition of a sense would be "a system that consists of a sensory cell type (or group of cell types) that respond to a specific kind of physical energy, and that correspond to a defined region (or group of regions) within the brain where the signals are received and interpreted". Where disputes arise is with regard to the exact classification of the various cell types and their mapping to regions of the brain.
There is no firm agreement amongst neurologists as to exactly how many senses there are; the disagreements stemming from a lack of a common consensus as to what the precise definition of a sense actually is. Although schoolchildren are routinely taught that there are five senses (sight, hearing, touch, smell, taste; a classification first devised by Aristotle), there is no doubt that this is most certainly in error. Regardless of the dispute, it is agreed that there are at least nine different senses in humans, and a minimum of two more observed in other organisms.
Using this definition several senses can be identified. This list begins with those five senses defined by Aristotle and hence probably most familiar to the reader.
Seeing or vision describes the ability to detect light and interpret it as "sight". There is disagreement as to whether or not this constitutes one, two or even three distinct senses. Neuroanatomists generally regard it as two senses, given that different receptors are responsible for the perception of colour (the frequency of light) and brightness (the energy of light). Some argue that the perception of depth also constitutes a sense, but it is generally regarded that this is really a cognitive (that is, post-sensory) function derived from having stereoscopic vision (two eyes) and is not a sensory perception as such.
Hearing or audition is the sense of sound perception and results from tiny hair fibres in the inner ear detecting the motion of atmospheric particles within (at best) a range of 20 to 20000 Hz. Sound can also be detected as vibration by tactition. Lower and higher frequencies than can be heard are detected this way only.
Taste or gustation is one of the two "chemical" senses. It is well-known that there are at least four types of taste "bud" (receptor) and hence, as should now be expected, there are anatomists who argue that these in fact constitute four or more different senses, given that each receptor conveys information to a slightly different region of the brain.
The four well-known receptors detect sweet, salt, sour, and bitter, although the receptors for sweet and bitter have not been conclusively identified. A fifth receptor called "umami" was first theorised in 1908 and its existence confirmed in 2000 (see  (http://www.nature.com/neuro/press_release/nn0200)). The umami receptor detects the amino acid glutamate, a flavor commonly found in meat, and in artificial flavourings such as monosodium glutamate.
Smell or olfaction is the other "chemical" sense. Olfactory neurons differ from most other neurons in that they die and regenerate on a regular basis.
The remaining senses can be considered types of physical feeling.
Tactition is the sense of pressure perception.
Thermoception is the sense of heat and the absence of heat (cold). It is also the first of the group of senses not identified explicitly by Aristotle. Again there is some disagreement about how many senses this actually represents--the thermoceptors in the skin are quite different from the homeostatic thermoceptors which provide feedback on internal body temperature. How warm or cold something feels does not only depend on temperature, but also on specific heat capacity and heat conductance; e.g., warm metal feels warmer than warm wood, and cold metal feels colder than cold wood, because metal has a higher thermal conductivity than wood. Wind feels cold because of the heat withdrawn for evaporation of sweat or other moisture, and because an isolating layer of warm air around the body blows away; however, in the case of hot air, wind makes it feel hotter, for a similar reason as the latter.
Nociception is the perception of pain. It can be classified as from one to three senses, depending on the classification method. The three types of pain receptors are cutaneous (skin), somatic (joints and bones) and visceral (body organs).
Equilibrioception is the perception of balance and is related to cavities containing fluid in the inner ear. There is some disagreement as to whether or not this also includes the sense of "direction" or orientation. However, as with depth perception earlier, it is generally regarded that "direction" is a post-sensory cognitive awareness.
Proprioception is the perception of body awareness and is a sense that people rely on enormously, yet are frequently not aware of. More easily demonstrated than explained, proprioception is the "unconscious" awareness of where the various regions of the body are located at any one time. (This can be demonstrated by anyone closing their eyes and waving their hand around. Assuming proper proprioceptive function, at no time will the person lose awareness of where the hand actually is, even though it is not being detected by any of the other senses).
Based on this outline and depending on the chosen method of classification, somewhere between 9 and 21 human senses have been identified. Additionally there are some other candidate physiological experiences which may or may not fall within the above classification, for example the sensory awareness of hunger and thirst.
Electroception (or "electroreception"), the most significant of the non-human senses, is the ability to detect electric fields[?]. Several species of fish, sharks and rays have evolved the capacity to sense changes in electric fields in their immediate vicinity. Some fish passively sense changing nearby electric fields, some generate their own weak, electric fields and sense the pattern of field potentials over their body surface, and some use these generating and sensing capacities for social communication. The mechanisms by which electroceptive fishes construct a spatial representation from very small differences in field potentials involve comparisons of spike latencies from different parts of the fish's body.
Magnetoception[?] (or "magnetoreception") is the ability to detect fluctuations in magnetic fields and is most commonly observed in birds. Although there is no dispute that this sense exists in many avians (it is essential to the navigational abilities of migratory birds) it is not a well understood phenomenon  (http://www.ks.uiuc.edu/Research/magsense/ms).
Echolocation is the ability to determine orientation to other objects through interpretation of reflected sound (like sonar). Bats and dolphins are noted for this ability, though some other mammals and birds do as well. It is most often used to navigate through poor lighting conditions or to identify and track prey. There is presently an uncertainty as to whether this is simply an extremely developed post-sensory interpretation of auditory perceptions, or actually constitutes a separate sense. Resolution of the issue will require brain scans of animals while they actually perform echolocation, a task which has proved difficult in practice.
[What about perception of polarization of light, by fish? Is there a one-word name for that?]