Humans have four main types of teeth. Incisors are the ones in front. Molars are the flat surfaced ones in back. The transitional teeth in front of the molars are called “premolars” or “bicuspids,” and the ones between these and the incisors are the canines – so named because their fang-like appearance can make them look a little like dogs’ teeth.
Canines in the upper arch are also often called “eye teeth,” which, when you come to think of it, is kind of a weird name for them. But the most common explanation is actually pretty simple: these teeth lie directly below the eye sockets.
Traditional Chinese Medicine also notes an energetic relationship between the canines and eyes. They lie on the same meridian – gall bladder/liver – which also includes the heart, pituitary gland, trunk musculature and joints of the feet, hips and knees.
According to Robert Hendrickson’s Facts on File Encyclopedia of Word and Phrase Origins, via a discussion on Phrase Finder,
The eyeteeth are those directly under the eye, the long, pointed canines that are cut at 14 to 20 months for the first set and at 11 to 12 years of age for the second set. To “cut one’s eyeteeth” means…”to acquire wisdom and become worldly,” because the permanent set is acquired when a child is passing into young adulthood. It is usually said in the negative, as in “he hasn’t cut his eyeteeth yet.” The expression was used by Haliburton’s Slick Sam in 1837 and Emerson after him, but is British in origin, dating back to the early 1700s, when it was “to have one’s eyeteeth.” “Eyeteeth” commonly referred to the canine teeth of dogs and other animals long before this, so the phrase may have been suggested by the fact that fighting dogs were considered dangerous to handle when they developed their eyeteeth. Actually, the words better described the emergence from infancy or childhood than they do the acquiring of wisdom.
Word expert Michael Quinion’s take is similar as he unpacks another common phrase: to “give one’s eye teeth for” something:
Why people seize on eye teeth as a dramatic way to indicate their longing for something is harder to get a grip on. If only you were asking about cut one’s eye teeth or cut one’s teeth, I could respond at once by pointing out that the eye teeth are among the last of a baby’s first set of teeth to appear and so to cut them (have them emerge from the gums) implies that babyhood is in effect over. To say that somebody has cut his eye teeth means he’s wide awake and isn’t easily fooled. If you’re cutting your eye teeth (or just teeth) on something you’re gaining experience in a situation you’re new to.
These suggest that eye teeth are especially valuable, because they figuratively embody hard-learned skills and one’s experience of life. The association with eyes results in an even more powerful evocation. To lose them would cause one to be severely hampered, not merely in eating but in everyday affairs.