The Pali word here is "tanha", meaning desire, wanting, or thirst. I prefer thirst, because we've all been thirsty. It is a base, animal urge for something. Thirst captures it because it is an urge we feel very physically, in our bodies.
The problem is that what we crave doesn't last. It doesn't last because everything is impermanent. Because it doesn't last, is cannot satisfy. In fact, craving is at its most intense when the thing we want isn't there. Once we have something, after a short period of relief, we usually take it for granted and move on to craving the next thing.
Non-clinging, or non-attachment, is the golden thread that links classic, Pali Buddhism with Zen. No one does non-attachment like Zen. "If you meet the Buddha on the road, kill him!"
In the classic Heart Sutra, it says:
Therefore, in emptiness there is no form, feeling, cognition, formation, or consciousness; no eyes, ears, nose, tongue, body, or mind; no sights, sounds, smells, tastes, objects of touch, or dharmas; no field of the eyes, up to and including no field of mind-consciousness; and no ignorance or ending of ignorance, up to and including no old age and death or ending of old age and death. There is no suffering, no accumulation, no cessation, no Way. And no understanding and no attaining. Because nothing is attained, the Bodhisattva, through reliance on prajna paramita, is unimpeded in his mind. Because there is no impediment, he is not afraid, and he leaves distorted dream-thinking far behind. Ultimately Nirvana!
The Heart Sutra negates every classic distinction of Pali Buddhism. It negates the Five Skandhas, the Four Noble Truths, the 12 Links of Dependent Origination. When everything is negated, where is there left to stand?
Hui Neng was enlightened hearing the words of the Diamond Sutra, "You should activate the mind without dwelling on anything."
Zen Master Powha Sunim used to quote the Bible here: Foxes have holes, birds have thier nests, but the Son of Man has nowhere to lay His head on it."
Uchiyama Roshi used to say, "Open the hand of thought."
One difficult thing about spiritual practice is this deep tendency to cling and crave to things. In fact, we often trade one clinging for another. Becoming interested in Buddhism, we may cling to Buddhism. Learning about mindfulness and attention, we may cling to mindfulness and attention. Learning concentration, we may cling to concentrate. One after another, the clinging remains, only the object changes. Many people believe this to be spiritual progress, but in fact they are only reinforcing this tendency to cling and crave.
Perhaps the worst clinging and craving has to do with the "Big Spiritual Experience." The "Big Spiritual Experience", or the "Cosmic Orgasm", is often see as the goal of spiritual practice and the sign of enlightenment. This would include "Becoming One with the Universe," "Everything Making Sense," or other such terms.
These experiences don't last. But the clinging and craving does. I think it is important to keep this in mind, even as we discover the wonders of Buddhism, Taoism, mindfulness, or concentration that, while useful, none of these things are the end all be all of spiritual practice.
The key motto of Zen is: Walk On.
Heart Sutra translated by Ron Epstein