Attitude of Mind
The question of what we should be doing with our mind during zazen (or any type of meditation for that matter) is answered differently in different traditions and by different teachers. There are two general approaches within Buddhism; in this Zen tradition they are referred to as concentration and observation. Individuals often prefer one over the other, but they are best viewed as complementary practices. The commonality between the two is paying attention; in fact, a succinct instruction is, "Shut up, sit down, and pay attention!" The difference is in what one pays attention to.
When practicing concentration, focus your attention on something predetermined—in the case of this tradition, the posture and the breath—and return your attention there whenever you become aware that your mind has wandered. You do this over and over and over. Concentration practice has an element of will in it: you are directing your mind, your attention. When practicing observation, attention is paid to whatever is presently occurring in the body/mind, including physical sensations, internalized words, images, feelings, desires—whatever is actually happening. Through observation you can become intimate with your own body/mind, discover what it actually does, how it actually works, and what it actually is.
As stated earlier, these practices are not incompatible. Most people discover that their mind is capable of doing more than one thing at a time, so it is usually not so difficult to pay attention to the breath and at the same time be aware that all sorts of other phenomena are going on: sensations, thoughts, fantasies, feelings, etc. In fact, when they first begin to sit, most people report the mind being something of a jumble, a condition classically described as "monkey mind" for its hyperkinetic activity. In part, especially in the beginning of practice, zazen is about the calming of this hyperactive mind. Some people find that concentration is a great aid in doing this; for others simply watching the monkey jump around and letting him calm down on his own is more effective. Most people use a combination of both.
Some teachers emphasize concentration practice as a way of facilitating samadhi, a condition usually described as a state of "non-dual consciousness," and given considerable importance in some traditions. In the Deshimaru tradition this is called hishiryo, a word that translates as "non-thinking" or "beyond thinking." Livingston-Roshi has called it "the unconscious of Zen." Terms like samadhi and hishiryo are problematic to talk about, for they, as non-dual "experiences," cannot be accurately described in dualistic language, and all language is of necessity dualistic. They are phenomena to be experienced rather than concepts to be grasped intellectually and accurately described verbally. They, like satori, kensho, or some other kind of special experience, are also problematic in that having an intention to seek, run after, try to have, hold on to, or repeat such "special" experiences can become just another form of craving, and craving lies at the root of suffering. In this tradition such deep states of consciousness, as well as sudden flashes of profound insight, may be experienced, but they are not sought, nor are they held on to should they occur. Like all phenomena, states of consciousness come and go, and our practice is more about waking up to their coming and going (which is the same as our coming and going) than it is about trying to have or hold on to any particular experience, profound as it might be.
As stated earlier, Dogen said that shikantaza, just-sitting, was the proper attitude of mind for a Zen practitioner during zazen. In shikantaza, we "just sit" wholeheartedly in the posture of zazen, paying mindful attention to whatever is occurring, letting it come and go. If that is monkey mind, we let that be there; if it's a profound state of samadhi, we let that be there. We just continue to sit wholeheartedly in the posture of zazen. Deshimaru talked about this attitude of being one of mushotoku, an attitude of "non-profit." We are not trying to gain anything personally from zazen. Kodo Sawaki famously said that "zazen is good for nothing."
No description of zazen can do it justice. You will never be able to know what zazen is simply by reading about it. That can only become known by doing it repeatedly, by actually putting your butt on the zafu and facing the wall.